Chapter 5.5, Problem 46E

To determine

To evaluate: The definite integral.

Answer to Problem 46E

The integration of the definite integral ${\displaystyle \underset{\frac{1}{6}}{\overset{\frac{1}{2}}{\int }}\csc \left(\pi t\right)\cot \left(\pi t\right)}dt$is $\frac{1}{\pi }$ .

Explanation of Solution

Given information: ${\displaystyle \underset{\frac{1}{6}}{\overset{\frac{1}{2}}{\int }}\csc \left(\pi t\right)\cot \left(\pi t\right)}dt$

Calculation: ${\displaystyle \underset{\frac{1}{6}}{\overset{\frac{1}{2}}{\int }}\csc \left(\pi t\right)\cot \left(\pi t\right)}dt$

Let, $u=\csc \left(\pi t\right)$

Differentiating with respect to t,

  $\begin{array}{l}\Rightarrow du=-\pi \cot \left(\pi t\right)\csc \left(\pi t\right)dt\\ \Rightarrow -\frac{du}{\pi }=\cot \left(\pi t\right)\csc \left(\pi t\right)dt\end{array}$

Transforming limits,

Upper limit, $\csc \left(\frac{\pi }{2}\right)\to 1$

Lower limit, $\csc \left(\frac{\pi }{6}\right)\to 2$

Let’s evaluate the definite integral,

  $\begin{array}{l}={\displaystyle \underset{2}{\overset{1}{\int }}-\frac{du}{\pi }}\\ =\frac{1}{\pi }{\displaystyle \underset{1}{\overset{2}{\int }}du},\text{ Taking constant out}\end{array}$

Applying the integral of a constant,

  $\begin{array}{l}=\frac{1}{\pi }{\left[u\right]}_1^2\\ =\frac{1}{\pi }\left(2-1\right)\\ =\frac{1}{\pi }\end{array}$

Hence, The integration of the definite integral ${\displaystyle \underset{\frac{1}{6}}{\overset{\frac{1}{2}}{\int }}\csc \left(\pi t\right)\cot \left(\pi t\right)}dt$is $\frac{1}{\pi }$ .