Question

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

Answer 1

`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`.