Integration by parts with trig functions

Evaluate $\displaystyle \int tan^{-1} \, \sqrt{x} \, dx$

For an integral involving the inverse tangent function, such as this one, the key idea is to use integration by parts, which is a powerful technique when you're dealing with the product of functions. Here, you’ll want to break the integral down into two parts: one function that can be easily differentiated, and another that can be integrated.

In this case, the inverse tangent function, which can be tricky to integrate directly, is a good candidate to differentiate. The square root of x, on the other hand, is easier to integrate, making it a natural choice for that part of the process.

After applying the integration by parts formula, you'll simplify the resulting expressions and work through the remaining integrals. In some cases, this might involve repeating the integration by parts process or applying basic integration techniques to complete the solution.

This type of problem illustrates how integration by parts allows us to tackle more complicated integrals by breaking them into simpler, more familiar components, especially when one of the functions (like inverse tangent) doesn’t lend itself to straightforward integration.

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