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inte^(x)(1+x)sec^(2)(xe^(x))dx=f (x)+ ...

`inte^(x)(1+x)sec^(2)(xe^(x))dx=f (x)+` Constant , then f (x) is equal to

A

`cos(xe^(x))`

B

`sin(xe^(x))`

C

`2tan^(-1)(x)`

D

`tan(xe^(x))`

Text Solution

Verified by Experts

The correct Answer is:
D
Given, `inte^(x)(1+x).sec^(2)(xe^(x))dx=f(x)+" constant"`
Put `xe^(x)=t rArr (e^(x)+xe^(x))dx=dt`
Here, LHS `=intsec^(2) t dt=tant+" constant"`
`rArr tan(xe^(x))+"constant "=f(x)+" constant"`
`rArr" "f(x)=tan(xe^(x))`
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