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Evaluate: int(-a)^a\ sqrt((a-x)/(a+x))\ ...

Evaluate: `int_(-a)^a\ sqrt((a-x)/(a+x))\ dx\ \ `

A

`api`

B

`(api)/(2)`

C

`2api`

D

none of these

Text Solution

Verified by Experts

Put `x=acostheta` and `dx=-asintheta d theta`.
`[x=-aimpliestheta=pi]` and `[x=aimpliestheta=0]`.
`:.I=int_(pi)^(0)sqrt((1-costheta)/(1+costheta))*(-asintheta)d theta=aint_(0)^(pi){(2sin^(2)(theta//2))/(2cos^(2)(theta//2))}^(1//2)2sin((theta)/(2))cos((theta)/(2))d theta`
`=aint_(0)^(pi)2sin^(2).(theta)/(2)d theta=a int_(0)^(pi)(1-costheta)d theta=api`.
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