# Integral of e^(ax)*sin(x)

In this post we are going to integrate this seemingly hard integral, which in my opinion is actually very interesting!

When we have a product of a trigonometric and a non trigonometric function, the best technique to try out is integration by parts, which we are going to perform here. What we need to choose are *u* and *dv*, but in this case it doesn’t really matter.

As you may know,

Therefore,

This is pretty much the same integral we want to solve, except there is cos(*x*) instead of sin(*x*); this means the only thing we can do is perform integration by parts again:

Let’s expand this:

And now the interesting part: this integral is the same we had at the beginning, so we can write an equation and solve for the integral:

Finally,

Do you see anything familiar? If you know *Euler’s identity*,

you’ll notice that, if a = i,

This is in fact the condition in order for the integral to exist: *a* must not be equal to *i*. So if you find this integral (with *a* = *i*), you won’t have to solve it, because the answer is negative infinity.

Hope you liked this post, and if so, give it a like! If you have any questions leave a comment and I’ll be happy to help! The next post will be about the integral from 0 to infinity of the sinc function, i.e. sin(*x*)/*x*. Subscribe to stay updated!