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Provide Solution For  R.D.Sharma Maths Class 12 Chapter 18  Indefinite Integrals Exercise 18.25 Question 12 Maths Textbook Solution.

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Answer: -x \cot x+\ln |\sin x|+c

Hint: Here, using Integration by parts

         \int a b d x=a \int b d x-\int\left[\frac{d}{d x} a \int b d x\right] d x

Given: I=\int x \operatorname{cosec}^{2} x d x

Solution: I=\int x \operatorname{cosec}^{2} x d x

          I=x \int \operatorname{cosec}^{2} x-\int\left[\frac{d x}{d x} \cdot \int \cos e c^{2} x d x\right] d x

          \begin{aligned} &=x(-\cot x)-\int 1(-\cot x) d x \\ &=-x \cot x+\int \cot x d x \\ &=-x \cot x+\ln |\sin x|+c \end{aligned}

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