Question

Which of the following values represent the definite integral

$\intop\begin{matrix} \pi/4 \\ 0 \end{matrix}$ $e$ ^tanθ sec² θ dθ

1) e²

2) e - 1

3) e/2

4) e

Answer 1

The correct answer is

2) e - 1

Solution;

Apply integration by substitution as follows;

Let $u=tan(\theta) \implies du=sec^2(\theta)d\theta$

This gives a ne lower bound and upper bound as follows;

Lower bound $u=tan(0) =0$

Upper bound $u=tan({\pi\over 4}) =1$

$\implies \int_0^{\pi\over 4}e^{tan(\theta)}sec^2(\theta)d\theta=\int_0^1e^udu=[e^u]_0^1=e^1-e^0=e-1$