Integrating x times arccos x using Trigonometric Substitution

Find the indefinite integral of $\displaystyle x \arccos(x) \, dx$ using trigonometric substitution.

When faced with an integral that involves inverse trigonometric functions, trigonometric substitution can be a powerful tool. In particular, the problem here involves integrating the product of x and the inverse cosine of x, which calls for a strategic manipulation of the integral to simplify its evaluation. Trigonometric substitution is particularly useful because it allows us to transform the integral into a form that is often more manageable. This technique typically involves substituting a trigonometric function for the variable, thereby taking advantage of trigonometric identities to simplify the resulting expression.

In this problem, by substituting x with a trigonometric function, we benefit from the identity relationships of trigonometric and inverse trigonometric functions. One possible approach is to employ the substitution x = sin(theta), which inherently connects to arccos through identity transformations. From here, the integral can be addressed by transforming the differential dx and substituting in the expression for arccos(x). After making the appropriate substitutions, the integration step involves solving a trigonometric integral. This substitution not only facilitates the evaluation of the integral but also provides additional practice in manipulating and applying trigonometric identities and transformations effectively.

The complexity of this problem lies in correctly executing the substitution process and simplifying the resulting trigonometric integrals. Mastery of these techniques is not only crucial for solving this particular problem but also provides a foundation for tackling more complex integrals that involve combinations of polynomial, trigonometric, and inverse trigonometric functions.

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