Problem

(a) Use the definitions of hyperbolic functions in terms of exponentials to prove that

$$1-\operatorname{sech}^2x\equiv \tanh^2x$$ $$I_n=\int_0^{\frac13\ln2}\tanh^n(3x)\,dx \qquad n\in\mathbb Z,\ n\ge 0$$

(b) Show that

$$I_n=I_{n-2}-\frac{p^{\,n-1}}{3(n-1)} \qquad n\ge 2$$

where $p$ is a rational number to be determined.

(c) Hence determine the exact value of

$$\int_0^{\frac13\ln2}\tanh^5(3x)\,dx$$

giving your answer in the form $a\ln b+c$ where $a$, $b$ and $c$ are rational numbers to be found.