Chapter 7.2, Problem 57E

To determine

To determine the integral ${\displaystyle \underset{0}{\overset{1}{\int }}\frac{10\sqrt{\theta }}{{(1+{\theta }^{3/2})}^2}d\theta }$ by substitution.

Answer to Problem 57E

The value of the integral ${\displaystyle \underset{0}{\overset{1}{\int }}\frac{10\sqrt{\theta }}{{(1+{\theta }^{3/2})}^2}d\theta }$ is $\frac{10}{3}$

Explanation of Solution

Given information: ${\displaystyle \underset{0}{\overset{1}{\int }}\frac{10\sqrt{\theta }}{{(1+{\theta }^{3/2})}^2}d\theta }$

Formula used: ${\displaystyle \int x^{n}dx=\frac{x^{n+1}}{n+1}+C}$

Calculation : Put $1+{\theta }^{3/2}=t$ so that $\frac{3}{2}\sqrt{\theta }d\theta =dt$

Also, when $\theta =0;t=1$ and when $\theta =1;t=2$

The given integral becomes as:

  ${\displaystyle \underset{0}{\overset{1}{\int }}\frac{10\sqrt{\theta }}{{(1+{\theta }^{3/2})}^2}d\theta }=\frac{20}{3}{\displaystyle \underset{1}{\overset{2}{\int }}\frac{1}{t^2}dt}$

  $=-\frac{20}{3}{\left[\frac{1}{t}\right]}_1^2=-\frac{20}{3}\left[\frac{1}{2}-1\right]=\frac{10}{3}$

Hence, ${\displaystyle \underset{0}{\overset{1}{\int }}\frac{10\sqrt{\theta }}{{(1+{\theta }^{3/2})}^2}d\theta }=\frac{10}{3}$