Integration by Substitution Method

Integration by Substitution Method: Some Important Forms with Examples

Let f(x)  be an integrable function in one variable x. If we have to integrate some well-known twisted form of f(x),  then we may take the help of the substitution method. Here we list some important forms.

Form 1: $\int f(ax+b) dx$ can be solved by the substitution method. In this case, put $z=ax+b.$

For Example, find $\int (2x+1)^5 dx$.

Put $z=2x+1$.

Then $dz=2 dx \quad \Rightarrow dx=\frac{1}{2} dz$.

Now $\int (2x+1)^5 dx$ $=\int z^5 \cdot \frac{1}{2} dz$ $=\frac{1}{2} \int z^5 dz$

$=\frac{1}{2} \frac{z^{5+1}}{5+1} \quad$ [$\because \int x^n dx=\frac{x^{n+1}}{n+1}$]

$=\frac{(2x+1)^6}{12} \quad$ [$\because   z=2x+1$]

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