Application of Integration by Parts

Application of integration by parts

$\int f(x)g^{\prime }(x)dx=f(x)g(x)-\int f^{\prime }(x)g(x)dx$

Product of a polynomial and an exponential

Example 1

To Find $\int x{e}^{x}dx$

Let

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

then

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

Example 2

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

Let

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

then

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

by using the result from example 1

Example 3

To find $\int x^{5}n{e}^{2}xdx$

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)