Application of Integration by Parts

Application of integration by parts

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

Product of a polynomial and an exponential

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 $\int x^n e^x\, dx$

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)