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

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

A

`(1+sqrt(x))/((1-x)^(2))+c`

B

`(1+sqrt(x))/((1+x)^(2))+c`

C

`(1-sqrt(x))/((1-x)^(2))+c`

D

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

Text Solution

Verified by Experts

The correct Answer is:
D
Let `I=int(dx)/((1+sqrt(x))sqrt((x-x^(2))))`
If `sqrt(x)=sinp, " then " (1)/(2sqrt(x))dx=cos p dp`
` :. I=int (2sin p cos p dp)/((1+sinp)sinp cosp)`
`=2int (dp)/((1+sin p))`
`=2int((1-sinp)dp)/(cos^(2)p)`
`=2{int sec^(2)p dp-int(tanp sec p)dp}`
`=2(tanp-secp)+C`
`=2(sqrt((x)/((1-x)))-(1)/(sqrt((1-x))))+C=(2(sqrt(x)-1))/(sqrt((1-x)))+C`
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