Incredibly hard Integral, help wanted

Jun 2015
South Carolina
Hello everyone, I'm trying to solve an integral but it really looks hard. I've already tryied it by looking some examples over the internet but I cannot find any.

The point here is to attempt to solve the Integral by using all (or almost all) the types of integration: by parts, partial fractions, trig susbtitution, etc and point them out.


Where a=4 and b=11

I hope anyone have a solution with the steps included to analyze this mosnters integral

Thank you in advance for your help.
Last edited:
Sep 2012
Are you integrating all the terms in the bracket? You should place the dx in there so we know which terms we are integrating.
Jun 2015
South Carolina
Hi Sakonpure6, thank you for your interest, yeah dx is stated in the very last part of the integral tiny characters, look in the picture


As a I mentioned this Integral is pretty confusing but it is correct and it has a solution, according to the calculus teacher, I should mentioned that we didn't have any explanations or examples on how to proceed. "We are supposed to figure out this kind of integrals very easy"
Last edited:
Mar 2014
Hello. :)

Take it term by term. It's much less intimidating that way.

\(\displaystyle \int \sec^4(x)\tan^{11}(x)~dx\)

We have:

\(\displaystyle \dfrac{d}{dx} \sec(x) = \sec(x)\tan(x) \), and \(\displaystyle \tan^2(x) + 1 = \sec^2(x)\).

So, this leads us to try: \(\displaystyle u = \sec(x)\).

Then, \(\displaystyle du = \sec(x)\tan(x)~dx\) and we have:

\(\displaystyle \int u^3 (u^2- 1)^5~du\)

This still looks nasty, but we can evaluate using Pascal's triangle to give:

\(\displaystyle \int u^3 [u^{10} - 5u^8 + 10u^6 - 10u^4 + 5u^2 - 1] du\)

And you can take it from here. Since the last integral is indefinite, the result's probably going to look extremely unattractive.

A full solution would take a long time, especially to write out in latex. Could you show anything that you've been able to do?