Chapter 5.5, Problem 19E

To determine

To evaluate: The indefinite integral.

Answer to Problem 19E

The integration of the indefinite integral ${\displaystyle \int e^x\sqrt{1+e^x}}dx$is $\frac{2}{3}{\left(1+e^x\right)}^{\frac{3}{2}}+C$ .

Explanation of Solution

Given information: ${\displaystyle \int e^x\sqrt{1+e^x}}dx$

Calculation: ${\displaystyle \int e^x\sqrt{1+e^x}}dx$

Let apply the substitution $z=1+e^x$

Partially differentiate both side,

  $\Rightarrow dz=e^{x}dx$

Let’s evaluate the indefinite integral,

  $\begin{array}{l}={\displaystyle \int \sqrt{z}dz}\\ ={\displaystyle \int z^{\frac{1}{2}}dz}\end{array}$

Therefore, using the power rule,

  $\begin{array}{l}=\frac{z^{\frac{1}{2}+1}}{\frac{1}{2}+1}\\ =\frac{z^{\frac{3}{2}}}{\frac{3}{2}}\\ =\frac{2}{3}{z}^{\frac{3}{2}}\end{array}$

Now, Substitute back $z=1+e^x$ ,

  $=\frac{2}{3}{\left(1+e^x\right)}^{\frac{3}{2}}$

Adding a constant with the solution,

  $=\frac{2}{3}{\left(1+e^x\right)}^{\frac{3}{2}}+C$

Hence, The integration of the indefinite integral ${\displaystyle \int e^x\sqrt{1+e^x}}dx$is $\frac{2}{3}{\left(1+e^x\right)}^{\frac{3}{2}}+C$ .