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$$\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 xe^{x}\,dx$$\int xe^{x}\,dx$

Let

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

then

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

Let

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

then

$\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$$\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 $\int x^{5}ne^{2}x\,dx$$\int x^{5}ne^{2}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$$\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)