#### Please Solve R.D.Sharma class 12 Chapter 19 Definite Integrals Exercise 19.1 Question 11 Maths textbook Solution.

Answer: $\frac{1}{2}log2$

Hint: Use indefinite formula to solve the integral and then put the value of limit to get the required answer

Given: $\int_{\frac{\pi }{4}}^{\frac{\pi }{2}}\cot xdx$

Solution:$\int_{\frac{\pi }{4}}^{\frac{\pi }{2}}\cot xdx=\left [ log|\sin x| \right ]^{\frac{\pi }{2}}_{\frac{\pi }{4}}$                                        $\left ( \int \cot xdx=log|\sin x| \right )$

$=\left [ log|\sin \frac{\pi }{2} |-|\sin \frac{\pi }{4}|\right ]$

$=log1-log\frac{1}{\sqrt{2}}$                                                                                            $\left [ \sin \frac{\pi }{2} =1,\sin \frac{\pi }{4}=\frac{1}{\sqrt{2}}\right ]$

\begin{aligned} &=\log \frac{1}{\frac{1}{\sqrt{2}}} \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \quad\left[\log a-\log b=\log \frac{a}{b}\right] \\ &=\log \sqrt{2}=\log 2^{\frac{1}{2}} \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \;\quad\left[\log a^{m}=m \log a\right] \end{aligned}

$=\frac{1}{2}log2$