Question

What is `int e^xsec^2(e^x)dx`?

Answer 1

`inte^xsec^2(e^x)dx=tan(e^x)+"c"`

Explanation:

We need to evaluate `inte^xsec^2(e^x)dx`

Let `u=e^x` and `{du}/dx=e^x`. Then `dx=e^(-x)du` and

`inte^xsec^2(e^x)=inte^xsec^2 u e^(-x)du=intsec^2 udu=tanu+"c"=tan(e^x)+"c"`

Note: if you're not convinced of the last step, you can let `v=tanu` and work from there to prove it, but the FTC tells use that the step is indeed correct without needing to make any further substitutions