{"id":38947,"date":"2023-01-17T10:00:24","date_gmt":"2023-01-17T04:30:24","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=38947"},"modified":"2023-01-18T09:57:38","modified_gmt":"2023-01-18T04:27:38","slug":"plus-two-maths-chapter-wise-questions-answers-chapter-7","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/","title":{"rendered":"Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals"},"content":{"rendered":"

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals are part of Plus Two Maths Chapter Wise Questions and Answers<\/a>. Here we have given Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals.<\/p>\n\n\n\n\n\n\n\n\n\n\n
Board<\/strong><\/td>\nSCERT, Kerala<\/td>\n<\/tr>\n
Text Book<\/strong><\/td>\nNCERT Based<\/td>\n<\/tr>\n
Class<\/strong><\/td>\nPlus Two<\/td>\n<\/tr>\n
Subject<\/strong><\/td>\nMaths\u00a0Chapter Wise Questions<\/td>\n<\/tr>\n
Chapter<\/strong><\/td>\nChapter 7<\/td>\n<\/tr>\n
Chapter Name<\/strong><\/td>\nIntegrals<\/td>\n<\/tr>\n
Number of Questions Solved<\/strong><\/td>\n49<\/td>\n<\/tr>\n
Category<\/strong><\/td>\n Kerala Plus Two<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Kerala Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals<\/h2>\n

Plus Two Maths Integrals Three Mark Questions and Answers<\/h3>\n

Question 1.
\nIntegrate the following. (3 Score each)<\/p>\n

    \n
  1. \u222bsin x sin 2x sin 3x dx<\/li>\n
  2. \u222bsec2<\/sup>x cos2<\/sup>2x dx<\/li>\n<\/ol>\n

    Answer:
    \n1. We have sin x sin 2x sin 3x
    \n= 1\/2 (2 sin x sin 3x) sin 2x
    \n= 1\/2 (cos 2x – cos 4x) sin 2x
    \n= 1\/4 (2 sin 2x cos 2x – 2 cos 4x sin 2x)
    \n= 1\/4 [sin 4x – (sin 6x – sin 2x)]
    \n= 1\/4(sin 4x + sin 2x – sin 6x)
    \n\u222bsin x sin 2x sin 3x dx
    \n= \\(\\frac{1}{4}\\) \u222b(sin 4x + sin 2x – sin 6x) dx
    \n= –\\(\\frac{1}{16}\\) cos 4x – \\(\\frac{1}{8}\\) cos 2x + \\(\\frac{1}{24}\\) cos 6x + c.<\/p>\n

    2. sec2<\/sup>x cos2<\/sup>2x = \\(\\frac{\\left(2 \\cos ^{2} x-1\\right)^{2}}{\\cos ^{2} x}\\)
    \n= \\(\\left(\\frac{2 \\cos ^{2} x}{\\cos x}-\\frac{1}{\\cos x}\\right)^{2}\\) = (2cosx – secx)2<\/sup>
    \n= 4cos2<\/sup>x + sec2<\/sup>x – 4
    \n= 2(1 + cos2x) + sec2<\/sup>x – 4
    \n= 2cos2x + sec2<\/sup>x – 2
    \n\u222bsec2<\/sup> x cos2<\/sup> 2x dx = \u222b(2 cos 2x + sec2<\/sup> x – 2)dx
    \n= sin 2x + tan x – 2x + c.<\/p>\n

    Question 2.
    \nFind \\(\\int \\frac{2+\\sin 2 x}{1+\\cos 2 x} e^{x} d x\\)?
    \nAnswer:
    \n\"Plus
    \n= \u222bex<\/sup> [sec2<\/sup> x + tan x]dx
    \n= \u222bex<\/sup>[tanx + sec2<\/sup>x]dx = ex<\/sup> tanx + c.<\/p>\n

    Question 3.
    \nEvaluate \\(\\int \\frac{\\sec ^{2} x d x}{\\sqrt{\\tan ^{2} x+4}}\\)?
    \nAnswer:
    \nPut tanx = u, sec2<\/sup>xdx = dy
    \n\"Plus<\/p>\n

    Question 4.
    \nFind the following integrals.
    \n\"Plus
    \nAnswer:
    \n(i) I = \\(\\int_{0}^{\\frac{\\pi}{2}} \\frac{\\sin x}{1+\\cos ^{2} x} d x\\)
    \nPut cosx = t \u21d2 -sin xdx = dt
    \nWhen x = 0 \u21d2 t = cos0 = 1,
    \n\"Plus<\/p>\n

    (ii) I = \\(\\int_{0}^{1} x e^{x^{2}} d x\\)
    \nPut x2<\/sup> = t \u21d2 2xdx = dt
    \nWhen x = 0 \u21d2 t = 0,
    \nx = 1 \u21d2 t = 1
    \nI = \\(\\frac{1}{2} \\int_{0}^{1} e^{t} d t\\) =
    \n\"Plus
    \n= [e1<\/sup> – e0<\/sup>] = e – 1.
    \n\"Plus
    \nPut sin x = t \u21d2 cos xdx = dt
    \nWhen x = 0 \u21d2 t = sin0 = 0,
    \n\"Plus<\/p>\n

    (iv) I = \\(\\int_{0}^{2} x \\sqrt{x+2} d x\\)
    \nPut x + 2 = t2<\/sup> \u21d2 dx = 2tdt
    \nWhen x = 0 \u21d2 t = \\(\\sqrt{2}\\), x = 2 \u21d2 t = 2
    \n\"Plus<\/p>\n

    (v) I = \\(\\int_{0}^{\\frac{\\pi}{2}} \\sqrt{\\sin x} \\cos x d x\\)
    \nPut sin x = t \u21d2 cos xdx = dt
    \nWhen x = 0 \u21d2 t = sin0 = 0,
    \n\"Plus
    \n\"Plus
    \nPut tan x = t \u21d2 sec2<\/sup> xdx = dt
    \nWhen x = 0 \u21d2 t = tan 0 = 0,
    \n\"Plus<\/p>\n

    Question 5.
    \n(i) If f (x) is an odd function, then \\(\\int_{-a}^{a} f(x)\\) = ?
    \n(a) 0
    \n(b) 1
    \n(c) 2\\(\\int_{0}^{a} f(x)\\) dx
    \n(d) 2a
    \nEvaluate
    \n(ii) \\(\\int_{-\\pi \/ 2}^{\\pi \/ 2} \\sin ^{99} x \\cdot \\cos ^{100} x d x\\)
    \n(iii) \\(\\int_{-1}^{1} e^{|x|} d x\\)
    \nAnswer:
    \n(i) (a) 0.<\/p>\n

    (ii) Here, f(x) = sin99<\/sup>x.cos100<\/sup>x .then,
    \nf(-x) = sin99<\/sup>(- x).cos100<\/sup>(- x) = – sin99<\/sup> x. cos100<\/sup> x = -f(x)
    \n\u2234 odd function \u21d2 \\(\\int_{-\\pi \/ 2}^{\\pi \/ 2} \\sin ^{99} x \\cdot \\cos ^{100} x d x=0\\).<\/p>\n

    (iii) Here, f(x) = e|x|<\/sup>, f(-x) = e|-x|<\/sup> = e|x|<\/sup> = f(x)
    \n\u2234 even function.
    \n\"Plus
    \nwe have |x| = x, 0 \u2264 x \u2264 1
    \n\"Plus<\/p>\n

    Question 6.<\/p>\n

      \n
    1. Show that cos2<\/sup> x is an even function. (1)<\/li>\n
    2. Evaluate \\(\\int_{-\\pi \/ 4}^{\\pi \/ 4} \\cos ^{2} x d x\\) (2)<\/li>\n<\/ol>\n

      Answer:
      \n1. Let f(x) = cos2<\/sup>x \u21d2 f(-x) = cos2<\/sup> (-x) = cos2<\/sup> x = f(x) even.<\/p>\n

      2.
      \n\"Plus<\/p>\n

      Question 7.
      \nFind the following integrals.
      \n\"Plus
      \nAnswer:
      \n\"Plus<\/p>\n

      \"Plus<\/p>\n

      Question 8.
      \nFind the following integrals.
      \n\"Plus
      \nAnswer:
      \n\"Plus
      \n\"Plus
      \nAdd (1) and (2)
      \n\"Plus
      \n\"Plus<\/p>\n

      \"Plus<\/p>\n

      \"Plus<\/p>\n

      Question 9.
      \nFind the following integrals.<\/p>\n

        \n
      1. \\(\\int \\frac{1}{3+\\cos x} d x\\)<\/li>\n
      2. \\(\\int \\frac{2 x}{x^{2}+3 x+2} d x\\)<\/li>\n<\/ol>\n

        Answer:
        \n1. \\(\\int \\frac{1}{3+\\cos x} d x\\)
        \nPut t = tanx\/2 \u21d2 dt = 1\/2 sec2<\/sup> x\/2 dx
        \n\"Plus<\/p>\n

        2. \\(\\int \\frac{2 x}{x^{2}+3 x+2} d x\\) = \\(\\int \\frac{2 x}{(x+2)(x+1)} d x\\)
        \n\"Plus
        \n2x = A(x + 1) + B (x + 2)
        \nwhen x = -1, -2 = B ; B = -2
        \nwhen x = -2, -4 = -A ; A = 4
        \n\"Plus<\/p>\n

        = 4log(x + 2) – 2log (x + 1) + C.<\/p>\n

        Plus Two Maths Integrals Four Mark Questions and Answers<\/h3>\n

        Question 1.
        \nFind the following integrals.
        \n\"Plus
        \nAnswer:
        \n\"Plus
        \nx2<\/sup> + x +1 = A(x2<\/sup> + 1) + (Bx + C)(x + 2)
        \nPut x = -2 \u21d2 4 – 2 + 1 = 5A \u21d2 A = \\(\\frac{3}{5}\\)
        \nEquating the coefficients of x2<\/sup>
        \n\u21d2 1 = A + B \u21d2 B = 1 – \\(\\frac{3}{5}\\) = \\(\\frac{2}{5}\\)
        \nEquating the constants
        \n\u21d2 1 = A + 2C \u21d2 2C = 1 – \\(\\frac{3}{5}\\) = \\(\\frac{2}{5}\\) \u21d2 C = \\(\\frac{1}{5}\\)
        \n\"Plus<\/p>\n

        \"Plus
        \n\u21d2 1 = A(x – 1) + B(x + 3)
        \nPut x = 1 \u21d2 1 = 2A \u21d2 A = \\(\\frac{1}{2}\\)
        \nPut x = -3 \u21d2 1 = -4B \u21d2 B = – \\(\\frac{1}{4}\\)
        \n\"Plus<\/p>\n

        \"Plus
        \nEquating the constants; \u21d2 1 = A
        \nEquating the coefficients if t;
        \n\u21d2 0 = A + B \u21d2 B = -1
        \n\"Plus<\/p>\n

        Question 2.
        \nFind the following integrals.<\/p>\n

          \n
        1. \u222b e2x<\/sup> sin3xdx<\/li>\n
        2. \u222b x sin-1<\/sup>xdx<\/li>\n<\/ol>\n

          Answer:
          \n1. I = \u222be2x<\/sup> sin3xdx = \u222b sin 3x \u00d7 e2x<\/sup>dx
          \n\"Plus<\/p>\n

          \"Plus<\/p>\n

          2. \u222b x sin-1<\/sup>xdx = \u222b sin-1<\/sup>x \u00d7 xdx
          \n\"Plus<\/p>\n

          Question 3.
          \n(i) Which of the following is the value of \\(\\int \\frac{d x}{\\sqrt{a^{2}-x^{2}}}\\)? (1)
          \n\"Plus
          \n(ii) Evaluate \\(\\int \\frac{2 x}{x^{2}+3 x+2} d x\\) (3)
          \nAnswer:
          \n(i) [sin-1<\/sup>\\(\\frac{x}{a}\\) + c]<\/p>\n

          (ii)
          \n\"Plus
          \n\u21d2 2x = A(x + 1) + B(x + 2) \u21d2
          \nPut x = -2 and x = -1, we get A = 4, B = -2
          \n\"Plus<\/p>\n

          Question 4.<\/p>\n

            \n
          1. Choose the correct answer from the bracket.
            \n\u222bex<\/sup> dx = — (e2x<\/sup> + c, e-x<\/sup> + c, e2x<\/sup> + c) (1)<\/li>\n
          2. Evaluate: \u222b ex<\/sup> sin x dx<\/li>\n<\/ol>\n

            Answer:
            \n1. ex<\/sup> + c<\/p>\n

            2. I = \u222bex<\/sup> sinxdx = sinx.ex<\/sup> – \u222bcos x.ex<\/sup>dx
            \n= sin x.ex<\/sup> – (cos x.ex<\/sup> – \u222b(- sin x).ex<\/sup> dx)
            \n= sinx.ex<\/sup> – cosxex<\/sup> – \u222bsinx.ex<\/sup>dx
            \n= sin x.ex<\/sup> – cos xex<\/sup> – I
            \n2I = sin x.ex<\/sup> – cos xex<\/sup>
            \nI = \\(\\frac{1}{2}\\)ex<\/sup>(sinx – cosx) + c.<\/p>\n

            Question 5.
            \n(i) f(x)\u222bg(x) dx – \u222b(f'(x)\u222bg(x) dx)dx (1)
            \n(a) \u222bf'(x)g{x)dx
            \n(b) \u222bf(x)g'(x)dx
            \n(c) \u222b\\(\\frac{f(x)}{g(x)}\\)dx
            \n(d) \u222bf(x)g(x)dx
            \n(ii) Integrate sin-1<\/sup>\\(\\sqrt{\\frac{x}{a+x}}\\)dx w.r.to x. (3)
            \nAnswer:
            \n(i) (d) \u222bf(x)g(x)dx<\/p>\n

            (ii) \u222bsin-1<\/sup>\\(\\sqrt{\\frac{x}{a+x}}\\)dx,
            \nPut x = a tan2<\/sup>\u03b8, \u03b8 = tan-1<\/sup>\\(\\sqrt{\\frac{x}{a}}\\)
            \n\u21d2 dx = 2a tan\u03b8 sec2<\/sup>\u03b8 d\u03b8
            \nI = \u222bsin-1<\/sup>\\(\\left(\\frac{\\tan \\theta}{\\sec \\theta}\\right)\\) 2a tan\u03b8 sec2<\/sup>\u03b8 d\u03b8
            \n= \u222bsin-1<\/sup>(sin\u03b8)2a tan\u03b8 sec2<\/sup>\u03b8 d\u03b8
            \n= 2a\u222b\u03b8 tan\u03b8 sec2<\/sup>\u03b8 d\u03b8
            \nPut tan\u03b8 = t, \u03b8 = tan-1<\/sup> t \u21d2 sec2<\/sup>\u03b8 d\u03b8 = dt
            \n= 2a \u222b tan-1<\/sup> t (t) d\u03b8
            \n\"Plus
            \n= a[tan2<\/sup>\u03b8.\u03b8 – tan\u03b8 + \u03b8] + c
            \n= a[\u03b8(1 + tan2<\/sup>\u03b8) – tan\u03b8] + c
            \n\"Plus<\/p>\n

            Question 6.
            \nMatch the following. (4)
            \n\"Plus
            \nAnswer:
            \n\"Plus<\/p>\n

            Question 7.
            \nEvaluate \\(\\int \\frac{x}{\\sqrt{x+a}+\\sqrt{x+b}} d x\\)?
            \nAnswer:
            \n\"Plus<\/p>\n

            Question 8.
            \nMatch the following.
            \n\"Plus
            \nAnswer:
            \n1.
            \n\"Plus<\/p>\n

            2. \u222bsec x(sec x + tan x)dx = \u222b(sec2<\/sup> x + sec x. tan x)dx
            \n= tanx + secx + c.<\/p>\n

            3. \u222be3x<\/sup>dx = \\(\\frac{e^{3 x}}{3}\\) + c.<\/p>\n

            4. \u222b(sin x + cos x)dx = sin x – cosx + c.<\/p>\n

            Question 9.
            \nConsider the integral I = \\(\\int \\frac{x \\sin ^{-1} x}{\\sqrt{1-x^{2}}} d x\\)?<\/p>\n

              \n
            1. What substitution can be given for simplifying the above integral? (1)<\/li>\n
            2. Express I in terms of the above substitution. (1)<\/li>\n
            3. Evaluate I. (2)<\/li>\n<\/ol>\n

              Answer:
              \n1. Substitute sin-1<\/sup> x = t.<\/p>\n

              2. We have, sin-1<\/sup> x = t \u21d2 x = sint
              \nDifferentiating w.r.t. x; we get,
              \n\\(\\frac{1}{\\sqrt{1-x^{2}}}\\)dx = dt
              \n\u2234 I = \u222bt sin t dt.<\/p>\n

              3. I = \u222bt sin t dt = t.(-cost) -\u222b(-cost)dt = -t cost + sint + c
              \n= -sin-1<\/sup> x. cos (sin-1<\/sup> x) + sin(sin-1<\/sup> x) + c
              \nx – sin-1<\/sup> x.cos(sin-1<\/sup> x) + c.<\/p>\n

              Question 10.
              \nEvaluate \\(\\int_{0}^{\\pi \/ 4} \\log (\\tan x) d x\\).
              \nAnswer:
              \n\"Plus<\/p>\n

              Question 11.
              \nFind the following integrals.<\/p>\n

                \n
              1. \\(\\int \\frac{\\sec ^{2} x}{\\cos e c^{2} x} d x\\) (2)<\/li>\n
              2. \\(\\int \\frac{1}{x^{2}-6 x+13} d x\\) (2)<\/li>\n<\/ol>\n

                Answer:
                \n1. \\(\\int \\frac{\\sec ^{2} x}{\\cos e c^{2} x} d x\\) = \\(\\int \\frac{\\sin ^{2} x}{\\cos ^{2} x} d x\\) = \u222btan2<\/sup> xdx
                \n= \u222b(sec2<\/sup>x – 1)dx = tanx – x + c.<\/p>\n

                2. \\(\\int \\frac{1}{x^{2}-6 x+13} d x\\)
                \n\"Plus<\/p>\n

                Question 12.
                \nMatch the following. Justify your answer.
                \n\"Plus
                \nAnswer:
                \n\"Plus<\/p>\n

                Question 13.
                \n(i) \u222bsin2x dx = ? (1)
                \n(a) 2 cos x + c
                \n(b) -2 sin x + c
                \n(c) \\(\\frac{\\cos 2 x}{2}\\) + c
                \n(d) \\(-\\frac{\\cos 2 x}{2}\\) + c
                \n(ii) Evaluate \u222bex<\/sup> sin 2x dx (3)
                \nAnswer:
                \n(i) (d) \\(-\\frac{\\cos 2 x}{2}\\) + c.<\/p>\n

                (ii) Consider I = \u222bex<\/sup> sin 2x dx
                \n= \u222bsin 2x. ex<\/sup>dx = sinx.ex<\/sup> – 2\u222bcos 2x. ex<\/sup>dx
                \n= sin 2x.ex<\/sup> – 2 (cos 2x.ex<\/sup> + 2\u222bsin 2x. ex<\/sup>dx)
                \n= sin 2x. ex<\/sup> – 2 cos 2x ex<\/sup> – 4 \u222bsin 2x. ex<\/sup>dx
                \n= sin 2x. ex<\/sup> – 2 cos 2x ex<\/sup> – 4I
                \n5 I = sin 2x. ex<\/sup> – 2 cos 2x ex<\/sup>
                \nI = \\(\\frac{e^{x}}{5}\\) (sin 2x – 2 cos 2x).<\/p>\n

                Question 14.<\/p>\n

                  \n
                1. Resolve \\(\\frac{x^{2}+1}{x^{2}-5 x+6}\\) into partial fractions. (2)<\/li>\n
                2. Hence evaluate \u222b\\(\\frac{x^{2}+1}{x^{2}-5 x+6}\\). (2)<\/li>\n<\/ol>\n

                  Answer:
                  \n1.
                  \n\"Plus<\/p>\n

                  2.
                  \n\"Plus
                  \n5x – 5 = A(x – 2) + B(x – 3)
                  \nx = 2, 5 = -B, B = -5
                  \nx = 3, 10 = A, A = 10
                  \n(1) \u21d2 I = \u222b 1dx + \u222b\\(\\frac{10}{x-3}\\) dx – \u222b\\(\\frac{5}{x-2}\\) dx
                  \n= x + 10log(x – 3) – 5log(x – 2) + c.<\/p>\n

                  Question 15.
                  \nEvaluate \\(\\int_{0}^{4}\\) xdx as a limit of sum.
                  \nAnswer:
                  \nBy definition,
                  \n\\(\\int_{a}^{b}\\) f(x) dx =
                  \n(b – a)\\(\\lim _{n \\rightarrow \\infty} \\frac{1}{n}\\){f(a) + f(a + h) +…….+f(a + {n – 1)h)}
                  \nHere, a = 0, b = 4, f(x) = x, h = \\(\\frac{4-0}{n}=\\frac{4}{n}\\) \u21d2 nh = 4
                  \n\"Plus<\/p>\n

                  Question 16.<\/p>\n

                    \n
                  1. Define the real valued function f(x) = |x2<\/sup> + 2x – 3| (2)<\/li>\n
                  2. Evaluate \\(\\int_{0}^{2}\\)|x2<\/sup> + 2x – 3|dx. (2)<\/li>\n<\/ol>\n

                    Answer:
                    \n1. f(x) = |x2<\/sup> + 2x – 3| = |(x – 1) (x + 3)|
                    \nWe have;
                    \n\"Plus<\/p>\n

                    2. I = \\(\\int_{0}^{2}\\)|x2<\/sup> + 2x – 3|dx
                    \n\"Plus<\/p>\n

                    Question 17.
                    \nConsider the function f(x) = |x|+|x + 1|<\/p>\n

                      \n
                    1. Define the function f (x) in the interval [-2, 1]. (2)<\/li>\n
                    2. Find the integral \\(\\int_{-2}^{1}\\) f(x) dx (2)<\/li>\n<\/ol>\n

                      Answer:
                      \n1. Given, f(x) = |x|+|x + 1|.
                      \nWe have,
                      \n\"Plus
                      \nCombining these two functions, we get the function f(x).
                      \n\"Plus<\/p>\n

                      2.
                      \n\"Plus<\/p>\n

                      Question 18.
                      \nEvaluate \\(\\int_{\\sqrt{6}}^{\\sqrt{3}} \\frac{d x}{1+\\sqrt{\\tan x}} d x\\). (4)
                      \nAnswer:
                      \n\"Plus<\/p>\n

                      Plus Two Maths Integrals Six Mark Questions and Answers<\/h3>\n

                      Question 1.
                      \n(i) Fill in the blanks. (3)
                      \n(a) \u222b tan xdx = —
                      \n(b) \u222b cos xdx = —
                      \n(c) \u222b\\(\\frac{1}{x}\\)dx = —
                      \n(ii) Evaluate \u222bsin3<\/sup> xcos2<\/sup> xdx (3)
                      \nAnswer:
                      \n(i) (a) log|secx| + c
                      \n(b) sinx + c
                      \n(c) log|x| + c.<\/p>\n

                      (ii) \u222bsin3<\/sup> xcos2<\/sup> xdx = \u222bsin2<\/sup> xcos2<\/sup> x sin xdx
                      \n= \u222b(1 – cos2<\/sup> x)cos2<\/sup> x sin xdx
                      \nPut cos x = t \u21d2 – sin xdx = dt
                      \n\u2234 \u222b(1 – cos2<\/sup> x)cos2<\/sup> xsin xdx = -\u222b(1 – t2<\/sup> )t2<\/sup>dt
                      \n= \u222b(t4<\/sup> – t2<\/sup>)dt = \\(\\frac{t^{5}}{5}-\\frac{t^{3}}{3}\\) + c
                      \n= \\(\\frac{\\cos ^{5} x}{5}-\\frac{\\cos ^{3} x}{3}\\) + c.<\/p>\n

                      Question 2.
                      \nFind the following integrals.
                      \n\"Plus
                      \nAnswer:
                      \n(i) I = \u222b(3x – 2)\\(\\sqrt{x^{2}+x+1} d x\\)
                      \nLet 3x – 2 = A(2x + 1) + B
                      \n\u21d2 3 = 2 A \u21d2 A = \\(\\frac{3}{2}\\)
                      \n\u21d2 -2 = A + B \u21d2 -2 = \\(\\frac{3}{2}\\) + B
                      \n\u21d2 B = -2 – \\(\\frac{3}{2}\\) = – \\(\\frac{7}{2}\\)
                      \n\"Plus<\/p>\n

                      \"Plus
                      \nUsing (2) and (3) in (1) we have;
                      \n\"Plus<\/p>\n

                      (ii) I = \\(\\int \\frac{2 x-3}{x^{2}+3 x-18} d x\\)
                      \nLet 2x – 3 = A(2x + 3) + B
                      \n\u21d2 2 = 2A \u21d2 A = 1
                      \n\u21d2 -3 = 3A + B \u21d2 -3 = 3 + B \u21d2 B = -6
                      \n\"Plus<\/p>\n

                      \"Plus<\/p>\n

                      (iii) I = \\(\\int \\frac{5 x+2}{1+2 x+3 x^{2}} d x\\)
                      \nLet 5x + 2 = A{6x + 2) + B
                      \n\u21d2 5 = 6 A \u21d2 A = \\(\\frac{5}{6}\\)
                      \n\u21d2 2 = 2A + B \u21d2 2 = \\(\\frac{5}{3}\\) + B \u21d2 2 – \\(\\frac{5}{3}\\) = \\(\\frac{1}{3}\\)
                      \n\"Plus<\/p>\n

                      \"Plus<\/p>\n

                      (iv) I = \\(\\int \\frac{5 x+3}{\\sqrt{x^{2}+4 x+10}} d x\\)
                      \nLet 5x + 3 = A(2x + 4) + B
                      \n\u21d2 5 = 2A \u21d2 A = \\(\\frac{5}{2}\\)
                      \n\u21d2 3 = 4A + B \u21d2 3 = 10 + B \u21d2 B = -7
                      \n\"Plus<\/p>\n

                      \"Plus
                      \nUsing (2) and (3) in (1) we have;
                      \n\"Plus<\/p>\n

                      Question 3.
                      \nConsider the expression \\(\\frac{1}{x^{3}-1}\\)<\/p>\n

                        \n
                      1. Split it into partial fraction. (2)<\/li>\n
                      2. Evaluate \u222b \\(\\frac{1}{x^{3}-1}\\) dx (4)<\/li>\n<\/ol>\n

                        Answer:
                        \n1.
                        \n\"Plus
                        \n1 = A (x2<\/sup> + x + 1) + (Bx + c)(x + 1),
                        \nPut x = -1 \u21d2 1 = A(1 + 1 + 1) \u21d2 A= \\(\\frac{1}{3}\\)
                        \nEquating like terms.
                        \n0 = A + B \u21d2 B = – \\(\\frac{1}{3}\\), 1 = A + C \u21d2 C = \\(\\frac{2}{3}\\)
                        \n\"Plus<\/p>\n

                        2.
                        \n\"Plus
                        \nPut, x – 2 = D (2x – 1) + E ,
                        \n1 = 2 D \u21d2 D = \\(\\frac{1}{2}\\),
                        \n-2 = -D + E \u21d2 E = –\\(\\frac{3}{2}\\)
                        \n\"Plus<\/p>\n

                        \"Plus<\/p>\n

                        Question 4.
                        \n(i) Match the following (4)
                        \n\"Plus
                        \n(ii) Consider the function f(x) = \\(\\frac{x^{4}}{x+1}\\) Evaluate \u222bf(x)dx (2)
                        \nAnswer:
                        \n(i)
                        \n\"Plus<\/p>\n

                        (ii) Here the numerator is of degree 4 and denominator of degree 1. So to make it a proper fraction we have to divide Nr by Dr.
                        \n\"Plus<\/p>\n

                        Question 5.<\/p>\n

                          \n
                        1. Evaluate the as \\(\\int_{0}^{2}\\)x2<\/sup>dx the limit of a sum. (3)<\/li>\n
                        2. Hence evaluate \\(\\int_{-2}^{2}\\)x2<\/sup>dx (1)<\/li>\n
                        3. If \\(\\int_{0}^{2}\\) f(x)dx = 5 and \\(\\int_{-2}^{2}\\) f(x)dx = 0, then \\(\\int_{-2}^{0}\\) f(x)dx = …….. (2)<\/li>\n<\/ol>\n

                          Answer:
                          \n1. Here the function is f(x) = x2<\/sup>, a = 0, b = 2 and h = \\(\\frac{b-a}{n}=\\frac{2}{n}\\)
                          \n\\(\\int_{0}^{2}\\)x2<\/sup>dx =
                          \n\"Plus<\/p>\n

                          2. \\(\\int_{-2}^{2}\\) x2<\/sup>dx = 2 \\(\\int_{0}^{2}\\)x2<\/sup>dx = \\(\\frac{16}{3}\\)<\/p>\n

                          3.
                          \n\"Plus<\/p>\n

                          Question 6.
                          \nFind \u222b\\(\\sqrt{\\tan x}\\)xdx.
                          \nAnswer:
                          \nGiven;
                          \nI = \u222b\\(\\sqrt{\\tan x}\\)xdx,
                          \nPut tanx = t2<\/sup> \u21d2 sec2<\/sup>xdx = 2tdt \u21d2 dx = \\(\\frac{2 t d t}{1+t^{4}}\\)
                          \n\"Plus<\/p>\n

                          \"Plus<\/p>\n

                          \"Plus<\/p>\n

                          Question 7.
                          \n(i) Match the following. (2)
                          \n\"Plus
                          \n(ii) Integrate \\(\\frac{\\sec ^{2} x}{5 \\tan ^{2} x-12 \\tan x+14}\\) w.r.to x. (4)
                          \nAnswer:
                          \n(i)
                          \n\"Plus<\/p>\n

                          \"Plus<\/p>\n

                          \"Plus<\/p>\n

                          Question 8.<\/p>\n

                            \n
                          1. Evaluate \\(\\int_{0}^{1} \\sqrt{x} d x\\) (1)<\/li>\n
                          2. If \\(\\int_{0}^{a} \\sqrt{x} d x=2 a \\int_{0}^{\\pi \/ 2} \\sin ^{3} x d x\\), find the value of a. (3)<\/li>\n
                          3. Hence find \\(\\int_{a}^{a+1}\\)x dx. (2)<\/li>\n<\/ol>\n

                            Answer:
                            \n1.
                            \n\"Plus<\/p>\n

                            2. Given;
                            \n\"Plus<\/p>\n

                            3. When a = 0
                            \n\"Plus
                            \nWhen a = 4
                            \n\"Plus<\/p>\n

                            Question 9.
                            \n(i) Let f (x) be a function, then \\(\\int_{0}^{a}\\) f(x) dx = ? (1)
                            \n(a) 2 \\(\\int_{0}^{a}\\) f(x – a) dx
                            \n(b) \\(\\int_{0}^{a}\\) f(a – x) dx
                            \n(c) f(a)
                            \n(d) 2\\(\\int_{0}^{a}\\) f(a – x) dx
                            \nEvaluate
                            \n\"Plus
                            \nAnswer:
                            \n(i) (b) \\(\\int_{0}^{a}\\) f(a – x) dx<\/p>\n

                            (ii)
                            \n\"Plus
                            \n(1) + (2)
                            \n\"Plus
                            \n\u21d2 I = 1.<\/p>\n

                            \"Plus<\/p>\n

                            Question 10.
                            \nFind the following integrals.<\/p>\n

                              \n
                            1. \u222b\\(\\frac{2 e^{x}}{e^{3 x}-6 e^{2 x}+11 e^{x}-6} d x\\)<\/li>\n
                            2. \u222b\\(\\frac{(3 \\sin x-2) \\cos x}{5-\\cos ^{2} x-4 \\sin x} d x\\)<\/li>\n<\/ol>\n

                              Answer:
                              \n1.
                              \n\"Plus
                              \n\u21d2 1 = A(t – 2)(t – 3) + B(t – 1)(t – 3) + C(t – 1)(t – 2)
                              \nPut t = 1 \u21d2 1 = A(-1)(-2) \u21d2 A = \\(\\frac{1}{2}\\)
                              \nPut t = 2 \u21d2 1 = B(1)(-1) \u21d2 B = -1
                              \nPut t = 3 \u21d2 1 = B(2)(1) \u21d2 B = \\(\\frac{1}{2}\\)
                              \n\"Plus<\/p>\n

                              \"Plus<\/p>\n

                              2. I = \u222b\\(\\frac{(3 \\sin x-2) \\cos x}{5-\\cos ^{2} x-4 \\sin x} d x\\)dx
                              \nPut sin x = t \u21d2 cosxdx = dt
                              \n\"Plus
                              \n\u21d2 3t – 2 = A(t – 2) + B
                              \nEquating the coefficients if t; \u21d2 3 = A
                              \nEquating the constants
                              \n\u21d2 -2 = -2A + B \u21d2 -2 = -6 + B \u21d2 B = 4
                              \n\"Plus<\/p>\n

                              Question 11.<\/p>\n

                                \n
                              1. Find \u222b\\(\\frac{1}{x^{2}+a^{2}}\\)dx (1)<\/li>\n
                              2. Show that 3x + 1 = \\(\\frac{3}{4}\\)(4x – 2) + \\(\\frac{5}{2}\\) (2)<\/li>\n
                              3. Evaluate \\(\\int \\frac{3 x+1}{2 x^{2}-2 x+3} d x\\) (3)<\/li>\n<\/ol>\n

                                Answer:
                                \n1. \u222b\\(\\frac{1}{x^{2}+a^{2}}\\)dx = 1\/a tan-1<\/sup> x\/a + c.<\/p>\n

                                2. 3x + 1 = A \\(\\frac{d}{d x}\\)(2x2<\/sup> – 2x + 3) + B
                                \n= A(4x – 2) + B
                                \n3 = 4A; A = 3\/4
                                \n1 = -2A + B
                                \n1 = -3\/2 + B, B = 1 + 3\/2 = 5\/2
                                \n\u2234 3x + 1 = 3\/4(4x – 2) + 5\/2<\/p>\n

                                3.
                                \n\"Plus<\/p>\n

                                We hope the given Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals will help you. If you have any query regarding Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals, drop a comment below and we will get back to you at the earliest.<\/p>\n","protected":false},"excerpt":{"rendered":"

                                Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals are part of Plus Two Maths Chapter Wise Questions and Answers. Here we have given Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals. Board SCERT, Kerala Text Book NCERT Based Class Plus Two Subject Maths\u00a0Chapter Wise Questions Chapter Chapter 7 Chapter […]<\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[42728],"tags":[],"yoast_head":"\nPlus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals - A Plus Topper<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals\" \/>\n<meta property=\"og:description\" content=\"Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals are part of Plus Two Maths Chapter Wise Questions and Answers. Here we have given Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals. Board SCERT, Kerala Text Book NCERT Based Class Plus Two Subject Maths\u00a0Chapter Wise Questions Chapter Chapter 7 Chapter […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/\" \/>\n<meta property=\"og:site_name\" content=\"A Plus Topper\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/aplustopper\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-01-17T04:30:24+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2023-01-18T04:27:38+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/03\/Plus-Two-Maths-Chapter-Wise-Questions-and-Answers-Chapter-7-Integrals-3M-Q2.jpg\" \/>\n<meta name=\"twitter:card\" content=\"summary\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Kalyan\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"12 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Organization\",\"@id\":\"https:\/\/www.aplustopper.com\/#organization\",\"name\":\"Aplus Topper\",\"url\":\"https:\/\/www.aplustopper.com\/\",\"sameAs\":[\"https:\/\/www.facebook.com\/aplustopper\/\"],\"logo\":{\"@type\":\"ImageObject\",\"@id\":\"https:\/\/www.aplustopper.com\/#logo\",\"inLanguage\":\"en-US\",\"url\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2018\/12\/Aplus_380x90-logo.jpg\",\"contentUrl\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2018\/12\/Aplus_380x90-logo.jpg\",\"width\":1585,\"height\":375,\"caption\":\"Aplus Topper\"},\"image\":{\"@id\":\"https:\/\/www.aplustopper.com\/#logo\"}},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/www.aplustopper.com\/#website\",\"url\":\"https:\/\/www.aplustopper.com\/\",\"name\":\"A Plus Topper\",\"description\":\"Improve your Grades\",\"publisher\":{\"@id\":\"https:\/\/www.aplustopper.com\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/www.aplustopper.com\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"ImageObject\",\"@id\":\"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/#primaryimage\",\"inLanguage\":\"en-US\",\"url\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/03\/Plus-Two-Maths-Chapter-Wise-Questions-and-Answers-Chapter-7-Integrals-3M-Q2.jpg\",\"contentUrl\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/03\/Plus-Two-Maths-Chapter-Wise-Questions-and-Answers-Chapter-7-Integrals-3M-Q2.jpg\",\"width\":332,\"height\":112,\"caption\":\"Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals 3M Q2\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/#webpage\",\"url\":\"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/\",\"name\":\"Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals - A Plus Topper\",\"isPartOf\":{\"@id\":\"https:\/\/www.aplustopper.com\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/#primaryimage\"},\"datePublished\":\"2023-01-17T04:30:24+00:00\",\"dateModified\":\"2023-01-18T04:27:38+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/www.aplustopper.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals\"}]},{\"@type\":\"Article\",\"@id\":\"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/#webpage\"},\"author\":{\"@id\":\"https:\/\/www.aplustopper.com\/#\/schema\/person\/b82e489789d83ccf01d664235804eb43\"},\"headline\":\"Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals\",\"datePublished\":\"2023-01-17T04:30:24+00:00\",\"dateModified\":\"2023-01-18T04:27:38+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/#webpage\"},\"wordCount\":2502,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/www.aplustopper.com\/#organization\"},\"image\":{\"@id\":\"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/03\/Plus-Two-Maths-Chapter-Wise-Questions-and-Answers-Chapter-7-Integrals-3M-Q2.jpg\",\"articleSection\":[\"HSSLiVE\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/#respond\"]}]},{\"@type\":\"Person\",\"@id\":\"https:\/\/www.aplustopper.com\/#\/schema\/person\/b82e489789d83ccf01d664235804eb43\",\"name\":\"Kalyan\",\"image\":{\"@type\":\"ImageObject\",\"@id\":\"https:\/\/www.aplustopper.com\/#personlogo\",\"inLanguage\":\"en-US\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/a08b17f948aaea19c0551b8ee4b91ca0?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/a08b17f948aaea19c0551b8ee4b91ca0?s=96&d=mm&r=g\",\"caption\":\"Kalyan\"},\"url\":\"https:\/\/www.aplustopper.com\/author\/kalyan\/\"}]}<\/script>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals - A Plus Topper","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/","og_locale":"en_US","og_type":"article","og_title":"Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals","og_description":"Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals are part of Plus Two Maths Chapter Wise Questions and Answers. Here we have given Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals. Board SCERT, Kerala Text Book NCERT Based Class Plus Two Subject Maths\u00a0Chapter Wise Questions Chapter Chapter 7 Chapter […]","og_url":"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/","og_site_name":"A Plus Topper","article_publisher":"https:\/\/www.facebook.com\/aplustopper\/","article_published_time":"2023-01-17T04:30:24+00:00","article_modified_time":"2023-01-18T04:27:38+00:00","og_image":[{"url":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/03\/Plus-Two-Maths-Chapter-Wise-Questions-and-Answers-Chapter-7-Integrals-3M-Q2.jpg"}],"twitter_card":"summary","twitter_misc":{"Written by":"Kalyan","Est. reading time":"12 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Organization","@id":"https:\/\/www.aplustopper.com\/#organization","name":"Aplus Topper","url":"https:\/\/www.aplustopper.com\/","sameAs":["https:\/\/www.facebook.com\/aplustopper\/"],"logo":{"@type":"ImageObject","@id":"https:\/\/www.aplustopper.com\/#logo","inLanguage":"en-US","url":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2018\/12\/Aplus_380x90-logo.jpg","contentUrl":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2018\/12\/Aplus_380x90-logo.jpg","width":1585,"height":375,"caption":"Aplus Topper"},"image":{"@id":"https:\/\/www.aplustopper.com\/#logo"}},{"@type":"WebSite","@id":"https:\/\/www.aplustopper.com\/#website","url":"https:\/\/www.aplustopper.com\/","name":"A Plus Topper","description":"Improve your Grades","publisher":{"@id":"https:\/\/www.aplustopper.com\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/www.aplustopper.com\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"ImageObject","@id":"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/#primaryimage","inLanguage":"en-US","url":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/03\/Plus-Two-Maths-Chapter-Wise-Questions-and-Answers-Chapter-7-Integrals-3M-Q2.jpg","contentUrl":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/03\/Plus-Two-Maths-Chapter-Wise-Questions-and-Answers-Chapter-7-Integrals-3M-Q2.jpg","width":332,"height":112,"caption":"Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals 3M Q2"},{"@type":"WebPage","@id":"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/#webpage","url":"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/","name":"Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals - A Plus Topper","isPartOf":{"@id":"https:\/\/www.aplustopper.com\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/#primaryimage"},"datePublished":"2023-01-17T04:30:24+00:00","dateModified":"2023-01-18T04:27:38+00:00","breadcrumb":{"@id":"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/www.aplustopper.com\/"},{"@type":"ListItem","position":2,"name":"Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals"}]},{"@type":"Article","@id":"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/#article","isPartOf":{"@id":"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/#webpage"},"author":{"@id":"https:\/\/www.aplustopper.com\/#\/schema\/person\/b82e489789d83ccf01d664235804eb43"},"headline":"Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals","datePublished":"2023-01-17T04:30:24+00:00","dateModified":"2023-01-18T04:27:38+00:00","mainEntityOfPage":{"@id":"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/#webpage"},"wordCount":2502,"commentCount":0,"publisher":{"@id":"https:\/\/www.aplustopper.com\/#organization"},"image":{"@id":"https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/#primaryimage"},"thumbnailUrl":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/03\/Plus-Two-Maths-Chapter-Wise-Questions-and-Answers-Chapter-7-Integrals-3M-Q2.jpg","articleSection":["HSSLiVE"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/www.aplustopper.com\/plus-two-maths-chapter-wise-questions-answers-chapter-7\/#respond"]}]},{"@type":"Person","@id":"https:\/\/www.aplustopper.com\/#\/schema\/person\/b82e489789d83ccf01d664235804eb43","name":"Kalyan","image":{"@type":"ImageObject","@id":"https:\/\/www.aplustopper.com\/#personlogo","inLanguage":"en-US","url":"https:\/\/secure.gravatar.com\/avatar\/a08b17f948aaea19c0551b8ee4b91ca0?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/a08b17f948aaea19c0551b8ee4b91ca0?s=96&d=mm&r=g","caption":"Kalyan"},"url":"https:\/\/www.aplustopper.com\/author\/kalyan\/"}]}},"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/posts\/38947"}],"collection":[{"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/comments?post=38947"}],"version-history":[{"count":1,"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/posts\/38947\/revisions"}],"predecessor-version":[{"id":155048,"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/posts\/38947\/revisions\/155048"}],"wp:attachment":[{"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/media?parent=38947"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/categories?post=38947"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/tags?post=38947"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}