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What is int ((x^(e -1) + e^(x -1))dx)/(x...

What is `int ((x^(e -1) + e^(x -1))dx)/(x^(e) + e^(x))` equal to ?

A

`(x^(2))/(2) + c`

B

In `(x + e) + c`

C

In `(x^(e) + e^(x)) + c`

D

`(1)/(e) " In " (x^(e) + e^(x)) + c`

Text Solution

Verified by Experts

The correct Answer is:
D
`int ((x^(e-1) + e^(x -1))dx)/(x^(e) + e^(x))`
Put `x^(e) + e^(x) = t rArr ex^(e -1) + e^(x) = (dt)/(dx)`
`rArr (ex^(e -1) + e^(x)) dx = dt`
`:. int ((x^(e -1) + e^(x -1))dx)/(x^(e) + e^(x)) = (1)/(e) int (e.x^(e -1) + e^(x))/(x^(e) + e^(x)) .dx`
`= (1)/(e) int (dt)/(t) = (1)/(e).ln t + c`
`= (1)/(e).ln (x^(e) + e^(x)) + C`
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