Full of Math Examples
FULL OF MATH EXAMPLES

# Integration by parts Examples

Here are the all examples in Integration by parts method.

Integration by parts is a method for solving integration problems.
By using property of integragtion of product of two function.
$\int u d v=u v-\int v d u$

## Example #1

#### Find the integration of $x \cos x d x$ using integration by parts.

Put $f(x)=x$ (first function) and $g(x)=\cos x($ second function) Then, integration by parts gives
$\int x \cos x d x=$$x \int \cos x d x-\int\left[\frac{d}{d x}(x) \int \cos x d x\right] d x =x \sin x-\int \sin x d x$$=x \sin x+\cos x+C$

## Example #2

#### Find the integration of $e^{x} \sin x$ using integration by parts method.

Take $e^{x}$ as the first function and $\sin x$ as second function.
Then, Integrating by parts gives us,
$\mathrm{I}=\int e^{x} \sin x d x $$=e^{x}(-\cos x)+\int e^{x} \cos x d x \\ =-e^{x} \cos x+\mathrm{I}_{1}(\text { say }) ....(1) Taking e^{x} and \cos x as the first and second functions, respectively, in \mathrm{I}_{1}, we get \mathrm{I}_{1}=e^{x} \sin x-\int e^{x} \sin x d x Substituting the value of \mathrm{I}_{1} in (1), we get \mathrm{I}=-e^{x} \cos x+e^{x} \sin x-\mathrm{I} or 2 \mathrm{I}=e^{x}(\sin x-\cos x) Finally, \mathrm{I}=\int e^{x} \sin x d x$$=\frac{e^{x}}{2}(\sin x-\cos x)+\mathrm{C}$

## Example #3

#### Find the integration of $x^{2} \sin x d x$ using integration by parts method.

$\int x^{2} \sin x d x$$=-x^{2} \cos x+\int 2 x \cos x d x$
$=-x^{2} \cos x+2 x \sin x-\int 2 \sin x d x$
$=-x^{2} \cos x+2 x \sin x+2 \cos x+C$

## Example #4

#### Find the integration of $x^{3} e^{-x^{2}} d x$ using integration by parts method.

Firstly , we rewrite integral as $I= \int x^{2} (x e^{-x^{2}}) d x$
$I=-\frac{1}{2} x^{2} e^{-x^{2}}-\int(-x) e^{-x^{2}} d x$
$=-\frac{1}{2} x^{2} e^{-x^{2}}-\frac{1}{2} e^{-x^{2}}+c$

## Example #5

#### Find the integration of $e^{a x} \cos b x d x$ using integration by parts method.

Integrating by parts, taking $e^{a x}$ as the first function, we find
$I=e^{a x}\left(\frac{\sin b x}{b}\right)-\int a e^{a x}\left(\frac{\sin b x}{b}\right) d x$

Taking integration by parts second time$I=e^{a x}\left(\frac{\sin b x}{b}\right)-a e^{a x}\left(\frac{-\cos b x}{b^{2}}\right)+\int a^{2} e^{a x}\left(\frac{-\cos b x}{b^{2}}\right) d x$integral on the RHS is just $-a^{2} / b^{2}$ times the original integral $I$ . Thus
$I=e^{a x}\left(\frac{1}{b} \sin b x+\frac{a}{b^{2}} \cos b x\right)-\frac{a^{2}}{b^{2}} I$
Rearranging and adding integration constant gives
$I=\frac{e^{a x}}{a^{2}+b^{2}}(b \sin b x+a \cos b x)+c$