`intsin^(-1)sqrt((x)/(a+x))dx` का मान ज्ञात कीजिए ।

Explanation

`int sin^(-1)sqrt((x)/(a+x))dx`
`" "{:("माना "x=a tan^(2)theta),(rArr" "dx=2atan theta.sec^(2)theta d theta):}`
`=intsin^(-1)sqrt((a tan^(2)theta)/(a+a tan^(2) theta)).2a tan theta sec^(2) theta sec^(2) theta d theta`
`=2aintsin^(-1)sqrt((atan^(2)theta)/(a+atan^(2)theta)).tan theta sec^(2) theta d theta `
`=2aintsin^(-1)(sin theta).tan theta sec^(2)theta d theta `
`=2a int theta.(tan theta sec^(2) theta)d theta`
`=2a[theta.int tan theta sec^(2) theta d theta-int{(d)/(d theta).theta. int tan theta sec^(2) theta d theta}d theta]`
`=2a[theta. (tan^(2) theta)/(2)-int1.(tan^(2)theta)/(2)d theta]`
`a[ theta. tan^(2) theta int (sec^(2) theta-1)d theta]`
`=a [theta tan^(2) theta- tan theta+theta]+c`
`=a[tan^(1)sqrt((x)/(a)).(x)/(a)-sqrt((x)/(a))+c]+c`
`=(a+x) tan^(-1)sqrt((x)/(a))-sqrt(ax)+c`