Application of Integration by Parts

Application of integration by parts

${\displaystyle \int f(x)g'(x)\,dx=f(x)g(x)-\int f'(x)g(x)\,dx}$

Product of a polynomial and an exponential

Example 1

To Find ${\displaystyle \int xe^{x}\,dx}$

Let

${\displaystyle f(x)=x}$ and ${\displaystyle g'(x)=e^{x}}$

then

${\displaystyle \int xe^{x}\,dx=xe^{x}-\int e^{x}\,dx=xe^{x}-e^{x}=(x-1)e^{x}+C}$

Example 2

To Find ${\displaystyle \int x^{2}e^{x}\,dx}$

Let

${\displaystyle f(x)=x^{2}}$ and ${\displaystyle g'(x)=e^{x}}$

then

${\displaystyle \int x^{2}e^{x}\,dx=x^{2}e^{x}-2\int xe^{x}\,dx=x^{2}e^{x}-2(x-1)e^{x}=(x^{2}-2x+2)e^{x}+C}$

by using the result from example 1

Example 3

To find ${\displaystyle \int x^{5}ne^{2}x\,dx}$

here n is an integer

By mathematical induction and using the above two examples

${\displaystyle \int x^{n}e^{x}\,dx=(x^{n}-{n}x^{n-1}+n(n-1)x^{n-2}-n(n-1)(n-2)x^{n-3}+...+(-1)^{n}n!)e^{x}+C}$

Anish27 23:28, 10 November 2011 (UTC)