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

Options

A. `2(sqrt((x)/sqrt(1-x))-(1)/(sqrt(1-x)))+c`
B. `2(sqrt((x)/sqrt(1-x))-(1)/(1-x))+c`
C. `2(sqrt((x)/(1-x))-(1)/(sqrt(1-x)))+c`
D. `2(sqrt((x)/(1-x))-(1)/(1-x))+c`

Answer

C

Explanation

`Let int(1)/((1+sqrtx)sqrt(x-1))dx`
Putting `x=sin^(2)t and dx=2 sin t cos t dt,` we get
`I=int(2sint cos tdt)/((1+sint)sqrt(sin^(2)t-sin^(4)t))`
`rArr" "I=2int(1)/(1+sin t)dt=2int(1-sint)/(cos^(2)t)dt`
`=2int(sec^(2)t-sect tan t)dt`
`=2(tan t-sec t)+c`
`rArr" "2(sqrt((x)/(1-x))-(1)/(sqrt(1-x)))+c`