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Evaluate: int((x-1)e^x)/((x+1)^3)dx...

Evaluate: `int((x-1)e^x)/((x+1)^3)dx`

Text Solution

Verified by Experts

The correct Answer is:
`(e^(x))/((x+1)^(2))+c`
Let `I=int((x-1)e^(x))/((x+1)^(3))dx`
`I=int{(x+1-2)/((x+1)^(3))}e^(x) dx=int{(1)/((x+1)^(2))-(2)/((x+1)^(3))}e^(x) dx`
`=int e^(x)(1)/((x+1)^(2))dx-2 int e^(x)*(1)/((x+1)^(3))dx`
Applying integration by parts,
`={(1)/((x+1)^(2))*e^(x)-int e^(x)*(-2)/((x+1)^(3))dx}-2int e^(x)*(1)/((x+1)^(3))dx=(e^(x))/((x+1)^(2))+c`
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