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int(dx)/((1+sqrt(x))sqrt((x-x^(2)))) is ...

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

A

`-2sqrt((1+sqrt(x))/(1-sqrt(x)))+C`

B

`-2sqrt((1-sqrt(x))/(1+sqrt(x)))+C`

C

`-sqrt((1+sqrt(x))/(1+sqrt(x)))+C`

D

`2sqrt((1+sqrt(x))/(1-sqrt(x)))+C`

Text Solution

Verified by Experts

Let `x=cos^(2)theta`. then `dx=-2sin thetacos theta d theta`.
`thereforeI=int(1)/((1+sqrt(x))sqrt(x-x^(2)))dx`
`rArrI=int(1)/((1+costheta)costhetasintheta)xx(-2sinthetacos theta) d theta`
`rArrI=-int"sec"^(2)(theta)/(2)d theta=-2"tan"(theta)/(2)+C`
`rArrI=-2sqrt((1-costheta)/(1+costheta))+C=-2sqrt((1-sqrt(x))/(1+sqrt(x)))+C`
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OBJECTIVE RD SHARMA ENGLISH-INDEFINITE INTEGRALS-Solved Example
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