Application of Integration by Parts

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Application of integration by parts[edit]

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

Product of a polynomial and an exponential[edit]

Example 1[edit]

To Find  \int x e^x\, dx


 f(x) = x and  g'(x) = e^x


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

Example 2[edit]

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


  f(x) = x^2 and  g'(x) = e^x


 \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[edit]

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)