Integration by Substitution

When a given function f(x) is neither in the simple from nor it can be integrated by more inspection, in that case we can use any one, or more of the following methods to evaluate the given integral.

1. Integration by Substitution
2. Integration by Parts.
3. Integration thorough Partial Fractions.

Here, we shall discuss how the various types of integrals can be reduced to the standard from the method of SUBSTITUTION.

Integration of a given function can be readily obtained by substitution of suitable variable viz. t, v, z etc.

in place of the given independent variable x.

Let us consider the integral, I = ∫ƒ(×)dx

Substituting x by h (t) we get dx/dt  = h’ (t) = dx = h’ (t) dt.

Now, by replacing the function we get,

Now, by replacing the functions we get.

I =∫ƒ(x)dx  = ∫[h’ (t)].dt

Hence, I = ∫[h(t)].h’ (t)dt

Some Important Integrals

1 =∫(ax+b) = 1/a ∫ƒ(t)].dt (where ax + b = t)

Proof :

I =∫ƒ(ax+b) dx

Let  (ax + b) = t then d/dx (ax+b) = dt/dx

= a = dt/dx, or dx = 1/a dt

∴ I =∫ƒ(t). 1/a dt = 1/a ∫ƒ(t)dt

Note. For the integrals of the types  ∫(ax+b)n dx,

∫n√(ax + b) dx and ∫ dx/ (ax+bⁿ) we can substitute ax + b by t.