Chapter 7, Problem 7RE

To determine

To evaluate the integral.

Answer to Problem 7RE

The required solution is , ${\displaystyle {\int }_0^{\pi /4}{e}^{\tan x}{\sec }^{2}x}dx=e-1$ .

Explanation of Solution

Given information: The equation is, ${\displaystyle {\int }_0^{\pi /4}{e}^{\tan x}{\sec }^{2}x}dx$ .

Concept used: Definite integrals,

The definite integral of $f\left(x\right)$ over $\left[a,b\right]$ , denoted by ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx,}$ is given by, ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx,}={\left[F\left(\text{b}\right)\right]}_a^b=F\left(b\right)-F\left(a\right)$

Calculation : The required solution is obtained as,

  $\begin{array}{l}{\displaystyle {\int }_0^{\pi /4}{e}^{\tan x}{\sec }^{2}x}dx\\ \text{ let }u=\tan xu=\tan \left(0\right)=0\\ du={\sec }^{2}xdxu=\tan \left(\frac{\pi }{4}\right)=1\\ ={\displaystyle {\int }_0^1{e}^{u}du}\\ ={\left[e^u\right]}_0^1\\ =e-1\end{array}$

Hence, the required solution is ${\displaystyle {\int }_0^{\pi /4}{e}^{\tan x}{\sec }^{2}x}dx=e-1$ .