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int (0)^(2) sqrt((2+x)/(2-x)) dx is equa...

` int _(0)^(2) sqrt((2+x)/(2-x))` dx is equal to

A

`pi+2`

B

`pi+(3)/(2)`

C

`pi+1`

D

`pi`

Text Solution

Verified by Experts

The correct Answer is:
A
Let ` l = int_(0)^(2) sqrt((2+x)/(2-x)) dx = int_(0)^(2) (2+x)/(sqrt(4-x^(2)))dx`
` = int_(0)^(2) 2/(sqrt(4-x^(2))) dx + int _(0)^(2) x/(sqrt(4-x^(2)))dx`
Put ` x= 2 sin theta rArr dx=2 cos theta d theta ` in the first integral and `4-x^(2) = t rArr -2xdx = dt ` in the second integral
` l = 2 int_(0)^(pi//2) (2 cos theta d)/(2 cos theta) + int_(4)^(0) - (dt)/(2sqrt(t))` ,
` = 2 int_(0)^(pi//2) d theta+1/2 int _(0)^(4) t^(-1//2) dt`
` = 2 [ theta] _(0)^(pi//2) +1/2 [ (t^(1//2))/(1//2) ]_(0)^(4) = pi+2`
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