Chapter 7.2, Problem 64E

To determine

To determine the integral ${\displaystyle \underset{\pi /4}{\overset{3\pi /4}{\int }}\cot xdx}$ by substitution.

Answer to Problem 64E

The value of the integral ${\displaystyle \underset{\pi /4}{\overset{3\pi /4}{\int }}\cot xdx}$ is $0$

Explanation of Solution

Given information: ${\displaystyle \underset{\pi /4}{\overset{3\pi /4}{\int }}\cot xdx}$

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

Calculation :

The given integral becomes as:

  ${\displaystyle \underset{\pi /4}{\overset{3\pi /4}{\int }}\cot xdx}={\displaystyle \underset{\pi /4}{\overset{3\pi /4}{\int }}\frac{\cos x}{\sin x}dx}$

Put $\sin x=t$ so that $\cos xdx=dt$

Also, when $x=\pi /4;t=1/\sqrt{2}$ and when $x=3\pi /4;t=1/\sqrt{2}$

The given integral becomes as:

  ${\displaystyle \underset{\pi /4}{\overset{3\pi /4}{\int }}\cot xdx}={\displaystyle \underset{1/\sqrt{2}}{\overset{1/\sqrt{2}}{\int }}\frac{1}{t}dt}=0$

Hence, ${\displaystyle \underset{\pi /4}{\overset{3\pi /4}{\int }}\cot xdx}=0$