Determine $\int_a^b \frac{f'(x)}{\cos^2(f(x))}e^{\tan(f(x))}dx$ with the fundamental theorem of calculus

Question

Determine $$\int_a^b \frac{f'(x)}{\cos^2(f(x))}e^{\tan(f(x))}dx$$ (for $\frac \pi 2+ k\pi \not\in f([a,b])$ for all $k \in \mathbb{Z}$) with the fundamental theorem of calculus.

How to do this? I'm familiar with integrals of explicitly given functions, but this is new to me.

Thanks in advance!

Answer 1

Remember (or prove) that $\tan'(x)=\frac{1}{\cos^2(x)}$. Make the substitution $u=\tan(f(x))$ to obtain $du=\frac{f'(x)}{\cos^2(f(x))}dx$. Therefore, you will come up with the integral $\int_{\tan(f(a))}^{\tan(f(b))} e^udu$, which is easily solved.

Answer 2

well using $u=f(x)$ we get $dx=\frac{du}{f'(x)}$. This would give us $\sec^2(u)e^{\tan(u)}du$ which is of the form $g'(u)e^{g(u)}du$ and so is easily solveable.