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int e^(sin^(-1)x)((log(e)x)/(sqrt(1-x^(2...

`int e^(sin^(-1)x)((log_(e)x)/(sqrt(1-x^(2)))+(1)/(x))dx` is equal to

A

`log_(e)x*e^(sin^(-1)x)+c`

B

`(e^(sin^(-1)x))/(x)+c`

C

`-log_(e)x*e^(sin^(-1)x)+c`

D

`e^(sin^(-1)x)(log_(e)x+(1)/(x))+c`

Text Solution

Verified by Experts

The correct Answer is:
A
`int e^(sin^(-1)x)((log_(e)x)/(sqrt(1-x^(2)))+(1)/(x))dx`
`=int e^(sin^(-1)x)(log_(e)x)/(sqrt(1-x^(2)))dx+int(e^("sin"^(-1)x))/(x)dx`
`=log_(e)x int (e^(sin^(-1)x))/(sqrt(1-x^(2)))dx=int((log_(e)x)'int(e^(sin^(-1)x))/(sqrt(1-x^(2))))dx+int (e^(sin^(-1)x))/(x)dx`
`=log_(e)x*e^(sin^(-1)x)-int(1)/(x)e^(sin^(-1)x)dx+int (e^(sin^(-1)x))/(x)dx`
`=log_(e)x*e^(sin^(-1)x)+c`
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