# 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.

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