[Math] The average value of the function $y=\tan(2x)$ over the interval $[0,\frac{\pi}{8}]$

Question

I was given the following question in a technology free exam. How would one go about solving this without the use of a calculator?

_{NB. I am currently in my last year of high school so please don't suggest crazy equations that contain so many greek symbols it could be a greek dictionary – Unless of course that is the only way to complete the question.}

The average value of the function $y=\tan(2x)$ over the interval $[0,\frac{\pi}{8}]$ is
A. $\frac{2}{\pi}\log_e({2})$
B. $\frac{\pi}{4}$
C. $\frac{1}{2}$
D. $\frac{4}{\pi}\log_e({2}$
E. $\frac{8}{\pi}$

Answer 1

The general formula for the average value of a function $f(x)$ over $a \le x \le b$ is $$f_{\text{avg}} = \dfrac{1}{b-a}\displaystyle\int_a^bf(x)\,dx$$

Here, you want the avgerage value of the function $f(x) = \tan 2x$ over $0 \le x \le \dfrac{\pi}{8}$.

Can you use the above formula to get the answer?

EDIT: Since you are having trouble integrating $\tan 2x$, notice that $\displaystyle\int \tan 2x\,dx = \int \dfrac{\sin 2x}{\cos 2x}\,dx$.

Now, make a substitution such as $u = \cos 2x$.