#### Explain solution rd sharma class 12 chapter Indefinite Integrals exercise 18.2 question 32

$-cot x-cosec x+c$

\begin{aligned} &\text { Hint: } \mathrm{U} \operatorname{sing} \int \cos e c^{2} x d x \text { and } \int \operatorname{cosec} x \cot x d x\\ &\text { Given: } \int \frac{\cos e c x}{\cos e c x-\cot x} d x\\ &\text { Solution: } I=\int \frac{\cos e c x}{\cos e c x-\cot x} d x\\ &\text { Multiply and divide by } \operatorname{cosec} x+\cot x\\ &I=\int \frac{\operatorname{cosec} x(\operatorname{cosec} x+\cot x)}{(\cos e c x-\cot x)(\operatorname{cosec} x+\cot x)} d x\\ &=\int \frac{\operatorname{cosec}^{2} x+\operatorname{cosec} x \cot x}{\left(\operatorname{cosec}^{2} x-\cot ^{2} x\right)} d x \quad\left[\begin{array}{l} (a-b)(a+b)=a^{2}-b^{2} \\ \operatorname{cosec}^{2} x-\cot ^{2} x=1 \end{array}\right]\\ &=\int \frac{\operatorname{cosec}^{2} x+\operatorname{cosec} x \cot x}{1} d x\\ &=\int \operatorname{cosec}^{2} x d x-\int \operatorname{cosec} x \cot x d x\\ &=-\cot x-\cos e c x+c \end{aligned}