Chapter 7.2, Problem 61E

To determine

To determine the integral ${\displaystyle \underset{0}{\overset{7}{\int }}\frac{dx}{x+2}}$ by substitution.

Answer to Problem 61E

The value of the integral ${\displaystyle \underset{0}{\overset{7}{\int }}\frac{dx}{x+2}}$ is $\ln (9/2)$

Explanation of Solution

Given information: ${\displaystyle \underset{0}{\overset{7}{\int }}\frac{dx}{x+2}}$

Formula used: ${\displaystyle \int \frac{1}{x}dx=\ln (x)+C}$

Calculation : Put $x+2=t$ so that $dx=dt$

Also, when $x=0;t=2$ and when $x=7;t=9$

The given integral becomes as:

  ${\displaystyle \underset{0}{\overset{7}{\int }}\frac{dx}{x+2}}={\displaystyle \underset{2}{\overset{9}{\int }}\frac{dt}{t}}={\left[\ln (t)\right]}_2^9=\left[\ln (9)-\ln (2)\right]=\ln (9/2)$

Hence, ${\displaystyle \underset{0}{\overset{7}{\int }}\frac{dx}{x+2}}=\ln (9/2)$