Question

How do you integrate `int e^x tan sqrtx dx` using integration by parts?

Answer 1

You have to apply the definition: `intudv=uv - intvdu`. Nonetheless, this integral does seem solvable analytically.

Explanation:

If you apply it once:

Choose: `dv=exp(x)dx`, and `u=tansqrt(x)`
Which gives you: `v=exp(x)`, and `du=sec^2sqrt(x)/(2sqrt(x))dx`

Thus:

`intexp(x)tansqrt(x)dx= exp(x)*sec^2sqrt(x)/(2sqrt(x)) - intexp(x)sec^2sqrt(x)/(2sqrt(x))dx`

Unfortunately, the more you go, the more complicate it gets, neither if you make `u=exp(x)`, and `dv=tansqrt(x)dx`, the exponential "locks" the integrals, making them cumbersome. Maybe this integral is no solvable by integration by part, I am not sure even analytically.

Learn more: https://en.wikipedia.org/wiki/Differentiation_of_trigonometric_functions, https://en.wikipedia.org/wiki/Lists_of_integrals, https://en.wikipedia.org/wiki/Integration_by_parts

Advanced

A possible way to solve it is by Taylor expansion, expand `tansqrt(x)`, which is the problematic part of the integral. I will try to add a second answer using this method.