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inte^(tanx)(sec^(2)x+sec^(3)x sinx)dx is...

`inte^(tanx)(sec^(2)x+sec^(3)x sinx)dx` is equal to

A

`sec x e^(tanx)+C`

B

`tanx e^(tanx)+C`

C

`e^(tanx)+tanx+C`

D

`(1+tanx)e^(tanx)+C`

Text Solution

Verified by Experts

The correct Answer is:
B
Let `l=int e^(tanx)(sec^(2)x+sec^(3)x sinx)dx`
`=inte^(tanx)(1+tanx)sec^(2)xdx`
Put `tan x= t rArr sec^(2)x dx=dt`
`l=int e^(t)(1+t)dt=e^(t)+intte^(t)dt`
`=e^(t)+te^(t)-e^(t)+C=te^(t)+C= tanx e^(tanx)+C`
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