What is the Integral of #tan^2(x)sec^2(x) dx#?

1 Answer
Jun 7, 2016

Answer:

#tan^3x/3+C#

Explanation:

When working with integrals of tangent and secant, it may not always be apparent what to do. Just remember that the derivative of #tanx# is #sec^2x# and the derivative of #secx# is #secxtanx#.

Here, notice that #sec^2x# is already in the integral, and all that remains is #tan^2x#. That is, we have #tanx# in squared form accompanied by its derivative, #sec^2x#. This integral is ripe for substitution!

In the integral #inttan^2xsec^2xdx#, let #u=tanx# and #du=sec^2xdx#.

This gives us #inttan^2xsec^2xdx=intu^2du#. Performing this integration yields #u^3/3+C#, and since #u=tanx#, this becomes #tan^3x/3+C#.