## Integration Rules and Formulas

### Integral of a Function

A function ϕ(x) is called a primitive or an antiderivative of a function f(x), if **?'(x) = f(x)**.

Let f(x) be a function. Then the collection of all its primitives is called the indefinite integral of f(x) and is denoted by **∫**f(x) dx.

Thus,

where ϕ(x) is primitive of f(x) and c is an arbitrary constant known as the constant of integration.

### Integration Rules

**Chain rule :**

**∫***u.v dx*=*uv*+ ……… + (–1)_{1}– u’v_{2}+ u”v_{3}– u”’v_{4}+ (–1)^{n–1}u^{n–1}v_{n}^{n}**∫**u^{n}.v_{n}dx

Where stands for*n*^{th}differential coefficient of*u*and stands for*n*^{th}integral of*v*.**Sum Rule**

**∫***(f + g) dx =***∫**f dx +**∫**g dx**Difference Rule**

**∫**(f – g) dx =**∫**f dx –**∫**g dx**Multiplication by constant**

**∫**cf(x) dx = c**∫**f(x) dx**Power Rule (n≠-1)**

**∫**x^{n}dx = x^{n+1}/(n+1) + C

### Fundamental Integration Formulae

In any of the fundamental integration formulae, if x is replaced by ax+b, then the same formulae is applicable but we must divide by coefficient of x or derivative of (ax+b) i.e., a. In general, if **∫**f(x) dx = ϕ(x) + c, then