Integration of x Sin Inverse x

In this tutorial we shall find the integral of x sine inverse of x, and solve this problem with the help of the integration by parts methods

The integral of x sine inverse of x is of the form

Here the first function is ${\sin ^{ - 1}}x$ and the second function is $x$.

Using the formula for integration by parts, we have

Using the formula above, equation (i) becomes

Using the integral formula $\int {\sqrt {{a^2} - {x^2}} } dx = \frac{{x\sqrt {{a^2} - {x^2}} }}{2} + \frac{{{a^2}}}{2}{\sin ^{ - 1}}\left( {\frac{x}{a}} \right) + c$, we have