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int e^(x^4) (x + x^3 +2x^5) e^(x^2) dx ...

`int e^(x^4) (x + x^3 +2x^5) e^(x^2) dx` is equal to

A

`(1)/(2)x e^(x^(2))e^(x^(4))+c`

B

`(1)/(2) x^(2)e^(x^(4))+c`

C

`(1)/(2) e^(x^(2))e^(x^(4))+c`

D

`(1)/(2)x^(2) e^(x^(2))e^(x^(4))+c`

Text Solution

Verified by Experts

The correct Answer is:
D
Putting ` x^(2)=t, `
`I=(1)/(2)int e^(t^(2))(1+t+2t^(2))e^(t)dt`
`=(1)/(2) int e^(t)[t e^(t^(2))+(e^(t^(2))+2t^(2)e^(t^(2)))]dt`
`=(1)/(2)int e^(t)[f(t)+f'(t)]dt=(1)/(2)e^(t)(t e^(t^(2)))+C " where " t=x^(2)`
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