Chapter 7, Problem 16RE

To determine

To evaluate the integral.

Answer to Problem 16RE

The required solution is, ${\displaystyle \int e^x\sec \left(e^x\right)dx}\text{ =}\ln \left|\sec \left(e^x\right)+\tan \left(e^x\right)\right|+C$ .

Explanation of Solution

Given information: The equation is, ${\displaystyle \int e^x\sec \left(e^x\right)dx}$ .

Concept used: Indefinite integral.

The family of all antiderivatives of a function $f\left(x\right)$ is a indefinite integral of $f$

With respect to $x$ and is denoted by ${\displaystyle \int f\left(x\right).}$

If F is an function such that $F\text{'}\left(x\right)=f\left(x\right)$ , then ${\displaystyle \int f\left(x\right)}dx=F\left(x\right)+C,$

Where C is an arbitrary constant, called the constant of integration.

Calculation : The required solution is obtained as,

  $\begin{array}{l}{\displaystyle \int e^x\sec \left(e^x\right)dx}\\ \text{ let }u=e^x\\ du=e^{x}dx\\ \text{ =}{\displaystyle \int \sec \left(u\right)}du\\ =\ln \left|\sec \left(u\right)+\tan \left(u\right)\right|+C\\ =\ln \left|\sec \left(e^x\right)+\tan \left(e^x\right)\right|+C\end{array}$

Hence, the required solution is, ${\displaystyle \int e^x\sec \left(e^x\right)dx}\text{ =}\ln \left|\sec \left(e^x\right)+\tan \left(e^x\right)\right|+C$ .