Difficult Trig Integrals

Evaluate $\displaystyle \int \frac{x \, sin^{-1}x}{\sqrt{1-x^2}} \, dx$

For this integral, the key to solving it lies in recognizing that the inverse sine function, often written as arcsin, is combined with a rational expression involving a square root. This type of problem typically calls for the integration by parts method, especially when you have a product of functions like this.

Integration by parts is useful here because one part of the expression, the inverse sine function, can be simplified when differentiated, while the other part, involving the rational term, becomes easier to handle when integrated. When applying integration by parts, you will first choose which part of the expression to differentiate and which part to integrate, then use the integration by parts formula to gradually simplify the integral.

This problem may also involve recognizing that some trigonometric identities related to inverse functions can be helpful. Once the parts are integrated and recombined, the final answer will emerge after completing the steps of the method and possibly applying algebraic simplifications. This example demonstrates the power of integration by parts in dealing with integrals that involve inverse trigonometric functions and algebraic terms.

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