Application of Integration by Parts

Application of integration by parts
$$ \int f(x)g'(x)\, dx = f(x)g(x) - \int f'(x)g(x)\, dx $$

Example 1
To Find $$ \int x e^x\, dx   $$

Let
 * $$ f(x) = x $$ and   $$ g'(x) = e^x $$

then
 * $$ \int x e^x\, dx

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

Example 2
To Find $$ \int x^2 e^x\, dx   $$

Let
 * $$ f(x) = x^2 $$  and  $$ g'(x) = e^x $$

then
 * $$ \int x^2 e^x\, dx = x^2e^x - 2\int xe^x \, dx = x^2e^x - 2(x-1)e^x

= (x^2-2x+2)e^x +C $$

by using the result from example 1

Example 3
To find $$ \int x^5n e^2x\, dx $$

here n is an integer

By mathematical induction and using the above two examples


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

Anish27 23:28, 10 November 2011 (UTC)