## Integration by Substitution

### Integration by Substitution

**(1) When integrand is a function i.e.,∫f[?(x)] ?'(x) dx:**

Here, we put ?(x)=t, so that ?'(x) dx=dt and in that case the integrand is reduced to ∫f(t) dt.

**(2) When integrand is the product of two factors such that one is the derivative of the others i.e., I = ∫f(x)f'(x) dx:**

In this case we put f(x)=t and convert it into a standard integral.

**(3) Integral of a function of the form f(ax + b):**

Here we put ax + b = t and convert it into standard integral. Obviously if ∫f(x) dx = ?(x) then ∫f(ax + b) dx = \(\frac { 1 }{ a }\) ?(ax + b) + c.

**(4) If integral of a function of the form \(\frac { f'(x) }{ f(x) }\)**

**(5) If integral of a function of the form \({ [f(x)] }^{ n }f'(x)\)**

**(6) If the integral of a function of the form \(\frac { f'(x) }{ \sqrt { f(x) } } \)**

**(7) Standard substitutions**

### Integration by Substitution Problems with Solutions

**1.**

**Solution:**

**2.**

**Solution:**

**3.**

**Solution:**

**4.**

**Solution:**

**5.**

**Solution:**