If `I=int (dx)/((a^(2)-b^(2)x^(2))^(3//2))`, then I equals

Options

A. ` (x)/(sqrt(a^(2)-b^(2)x^(2)))+C`
B. ` (x)/(a^(2)sqrt(a^(2)-b^(2)x^(2)))+C`
C. ` (a x)/(sqrt(a^(2)-b^(2)x^(2)))+C`
D. none of these

Answer

B

Explanation

Write `I=int (dx)/(x^(3)(a^(2)//x^(2)-b^(2))^(3//2))`
and put `a^(2)//x^(2)=t+b^(2)`, so that `(-2a^(2)//x^(3))dx=dt`
` :. I=int((-1//2a^(2))dt)/(t^(3//2))`
`= -(1)/(2a^(2))int t^(-3//2)dt=(1)/(a^(2)sqrt(t))+C`
`=(1)/(a^(2)(a^(2)//x^(2)-b^(2))^(1//2))+C`
`=(x)/(a^(2)(a^(2)-b^(2)x^(2))^(1//2))+C`