Problem

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Let

$$I_n = \int_0^{\pi/4} e^{4x}\tan^n x \, dx$$

(a) Use integration to prove that, for $n \ge 1$

$$I_{n+1} = \frac{k}{n} - \frac{4}{n} I_n - I_{n-1}$$

where $k$ is a constant to be determined.

(b) Determine the exact value of $I_0$

Given that $I_1 = 3.7002$ to 5 significant figures,

(c) use the answer to part (a) to determine the value of $I_4$ to 3 significant figures.