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

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