Question

Which of the following values represent the definite integral $\displaystyle{\int_{0}^{\pi/4}e^{\tan{\theta}}\sec^2{\theta}d\theta}$

1) $e ^2$

2) e - 1

3) e/2

4) e

Answer 1

$\displaystyle \text{Let $u = \tan\theta$, when $\theta = 0,\, u = 0$ and when $\theta = \frac\pi4, \, u = 1$ and } du = \sec^2\theta d \theta \\ \implies \int_{0}^{\frac \pi4} e^{\tan \theta} \sec^2 \theta \, d\theta = \int^1_0 e^u \, du = \left[e^u\right]^1_0 = e^1-e^0 \\=e-1$