Question

Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`

Evaluate:

`inte^(sin-1) x ((x+sqrt(1-x^2))/(sqrt(1-x^2)))dx`

Answer 1

Let I = `int e^(sin^-1x)[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]*dx`

= `int e^(sin^-1x) [x + sqrt(1 - x^2)]*(1)/sqrt(1 - x^2)*dx`

Put sin^–1 x = t

∴ `(1)/sqrt(1 - x^2) * dx` = dt

and x = sin t

∴ I = `int e^t [sin t + sqrt(1 - sin^2 t)]*dt`

= `int e^t [sin t + sqrt(cos^2t)]*dt`

= `int e^t(sin t + cos t)*dt`

Let f(t) = sin t
∴ f'(t) = cos t

∴ I = `int e^t[f(t) + f'(t)]*dt`

= e^t . f(t) + c
= e^t . sin t + c
= `e^(sin^(–1)x) * x + c`

= `x * e^(sin^(-1)x) + c`