Integration By Substitution – Formula and Examples

Here you will learn what is integration by substitution method class 12 with examples.

Let’s begin –

Integration By Substitution

The method of evaluating an integral by reducing it to standard form by a proper substitution is called integration by substitution.

If $\phi(x)$ is continuously differentiable function, then to evaluate integrals of the form

$\int$ $f(\phi(x))$ $\phi'(x)$ dx, we substitute $\phi(x)$ = t and $\phi'(x)$ dx = dt

This substitution reduces the above integral to $\int$ f(t) dt.

After evaluating this integral we substitute back the value of t.

Also Read : Integration Formulas for Class 12 – Indefinite Integration

Example : Prove that $\int$ sin(ax + b) dx = $-1\over a$ cos(ax + b) + C.

Solution : Let ax + b = t. Then, d(ax + b) = dt $\implies$ a dx = dt $\implies$ dx = $1\over a$ dt

Putting ax + b = t and dx = $1\over a$ dt, we get

$\int$ sin(ax + b) dx = $1\over a$ $\int$ sin t dt

= $-1\over a$ cos t + C

= $-1\over a$ cos(ax + b) + C

Example : Evaluate $\int$ $cos^2x\over {sin^2x + sinx}$ dx

Solution : I = $\int$ $(1-sin^2x)cosx\over {sinx(1 + sinx)}$ dx = $\int$ $1 – sinx\over {sinx}$ cosx dx

Put sinx = t  $\implies$  cosx dx = dt

$\implies$ I = $\int$ $1 – t\over t$ dt

= ln| t | – t + C

= ln|sinx| – sinx + C