Q. 44 - Calculus

Question

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

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

Step-by-Step Solution

Answer

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

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,