# Integral of sec²(x)/exp(tan(x))

Although it may look a bit intimidating at first, I can guarantee you this is actually a very easy integral!

But what integration technique do you think will be useful? You might think we could perform integration by parts, but I really don’t think it’s a good idea to find the integral of sec²(*x*) or 1/exp(tan(*x*)). What we need is u-substitution. The “hard” part is to choose what to substitute. If you remember, the derivative of tan(*x*) is sec²(*x*), which we have at the numerator, and all we have to do is simply substitute sec²(*x*)d*x* with d*u* and we will be left with 1/exp(*u*), which is way better!

Since sec²(*x*)d*x* is d*u* we will write

Let’s now talk about the function exp:

and, in general,

This means

We know that the integral of *e*^*u* is equal to *e*^*u*, so this integral is –*e*^-*u* (simply apply the chain rule); therefore

At this point we can undo the substitution (remember that *u*=tan(*x*)):

Overall:

That’s it! I really hope you liked this post and if so, give it a like! If you want to stay updated subscribe, it would mean a lot to me! Thanks!