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

In red: f(x)=e^(-ax)sin(x); in blue: F(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!

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