Question

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

${\int }_{\pi /4}^{\pi /2} \csc x\cot xdx$

Short answer

Ans: The exact value of,${\int }_{\pi /4}^{\pi /2} \csc x\cot xdx=\frac{1-\sqrt{2}}{\sqrt{2}}$

Step-by-step solution

Step 1. Given information.

given expression,

${\int }_{\pi /4}^{\pi /2} \csc x\cot xdx$

Step 2. The objective is to determine the exact value of the definite integral.   

The exact value is calculated as shown below,

$\begin{array}{r}{\int }_{\frac{\pi }{4}}^{\frac{\pi }{2}} \csc x\cot xdx\\ =[-\csc x{]}_{\frac{\pi }{4}}^{\frac{\pi }{2}}\\ =-\csc \frac{\pi }{2}+\csc \frac{\pi }{4}\\ =-1+\frac{1}{\sqrt{2}}\\ =\frac{1-\sqrt{2}}{\sqrt{2}}\end{array}$

Therefore, the exact value is$\frac{1-\sqrt{2}}{\sqrt{2}}$.

Step 3. Check:

The required graph is,