Consider the integral I = \(\int \frac{xsin^{-1}x}{\sqrt{1-x^2}}dx\)?

1.	Consider the integral I = \(\int \frac{xsin^{-1}x}{\sqrt{1-x^2}}dx\)?What substitution can be given for simplifying the above integral? Express I in terms of the above substitution.Evaluate I
Answer» 1. Substitute sin^-1 x = t. 2. We have, sin^-1 x = t ⇒ x = sint Differentiating w.r.t. x; we get, \(\frac{1}{\sqrt{1-x^2}}dx = dt\) ∴ I = ∫t sin t dt. 3. I = ∫t sin t dt = t.(-cost) -∫(-cost)dt = -t cost + sint + c = -sin^-1 x. cos (sin^-1 x) + sin(sin^-1 x) + c x – sin^-1 x.cos(sin^-1 x) + c.

Consider the integral I = \(\int \frac{xsin^{-1}x}{\sqrt{1-x^2}}dx\)?What substitution can be given for simplifying the above integral? Express I in terms of the above substitution.Evaluate I

Answer»

1. Substitute sin^-1 x = t.

2. We have, sin^-1 x = t ⇒ x = sint
Differentiating w.r.t. x; we get,

\(\frac{1}{\sqrt{1-x^2}}dx = dt\)

∴ I = ∫t sin t dt.

3. I = ∫t sin t dt = t.(-cost) -∫(-cost)dt = -t cost + sint + c
= -sin^-1 x. cos (sin^-1 x) + sin(sin^-1 x) + c
x – sin^-1 x.cos(sin^-1 x) + c.