Question

`int tan(sin^-1x)dx` = ______ 

Options

• `(1 - x^2)^{-1/2} + c`

• `(1 - x^2)^{1/2} + c`

• `tan^mx/sqrt(1 - x^2) + c`

• `-sqrt(1 - x^2) + c`

Answer 1

`int tan(sin^-1x)dx` = `underline(-sqrt(1 - x^2) + c)`

Explanation:

Let I = `int tan(sin^-1x)dx`

Put sin^-1x = t

⇒ x = sin t ⇒ dx = cos t . dt

∴ I = `int tan t. cos t. dt`

= `int (sin t)/(cos t) . cos t. dt`

= ∫sint . dt

= -cos t + c

= `-sqrt(1 - sin^2t) + c`

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