# How do you integrate int e^x sin sqrtx dx  using integration by parts?

May 11, 2018

I got overenthusiastic but I got stuck....I am not sure about it...I suspect it is either very complicated or not possible directly...

I got stuck...

May 11, 2018

You do not.

#### Explanation:

You are not going to find a satisfactory answer to this integral. That is, the result cannot be represented by elementary functions. For reference, an acceptable result of this integral would be:

$\int \left({e}^{x} \sin \left(\sqrt{x}\right)\right) \mathrm{dx} = - \left(\frac{1}{4}\right) i \left(\sqrt[4]{e} \sqrt{\pi} \text{ erf"(1/2 - isqrt(x)) - root4(e)sqrt(pi) " erf} \left(\frac{1}{2} + i \sqrt{x}\right) + 2 {e}^{x - i \sqrt{x}} \left(- 1 + {e}^{2 i \sqrt{x}}\right)\right) + C$

where $\text{erf} \left(x\right) = \frac{2}{\sqrt{\pi}} {\int}_{0}^{x} {e}^{- {t}^{2}} \mathrm{dt}$ and $i$ is the imaginary number.

You would not encounter this type of integral in a high school or college level calculus class. In fact, you would not see an integral of this type even while pursuing an undergraduate mathematics degree.