Integrals Involving Trig Functions

Evaluate$\displaystyle \int e^{cos^{-1}x}\, dx$

To solve an integral involving a combination of an exponential function and an inverse trigonometric function, like this one, you’ll want to approach the problem with substitution, which simplifies the structure of the integral.

In this case, the function inside the exponential is an inverse trigonometric function, specifically the inverse cosine. The substitution rule works by identifying a part of the expression that can be replaced with a simpler variable. By finding an appropriate substitution, you can convert the integral into a more familiar form—something easier to handle.

Once you apply the substitution, the integral transforms into something that is simpler to evaluate. After that, you'll integrate the new expression, and finally, reverse the substitution to express the answer in terms of the original variable.

This kind of problem highlights how substitution can simplify seemingly complex combinations of functions. Though the presence of inverse trigonometric and exponential functions can seem tricky at first, using the right substitution can break it down into manageable parts.

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