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  • Need Solution for R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise 18.25 Question 27 Maths Textbook Solution.

Need Solution for R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise 18.25 Question 27 Maths Textbook Solution.

Answers (1)

Answer: -\left[\sqrt{1-x^{2}} \cos ^{-1} x+x\right]+c

Given: \int \frac{x \cos ^{-1} x}{\sqrt{1-x^{2}}} d x

Hint: Use integration by parts

Solution:

            I=\int \frac{x \cos ^{-1} x}{\sqrt{1-x^{2}}} d x \Rightarrow-\frac{1}{2} \int \frac{-2 x}{\sqrt{1-x^{2}}} \cos ^{-1} x d x

[Using integration by parts]

              \begin{aligned} &=\frac{-1}{2}\left[\cos ^{-1} x \times \int \frac{-2 x}{\sqrt{1-x^{2}}} d x-\int\left\{\left(\frac{d}{d x} \cos ^{-1} x\right) \int \frac{-2 x}{\sqrt{1-x^{2}}} d x\right\} d x\right] \\ &=\frac{-1}{2}\left[\cos ^{-1} x \cdot 2 \sqrt{1-x^{2}}-\int-\frac{1}{\sqrt{1-x^{2}}} 2 \sqrt{1-x^{2}} d x\right] \\ &=\frac{-1}{2}\left[2 \sqrt{1-x^{2}} \cos ^{-1} x+2 x\right]+c \\ &=-\left[\sqrt{1-x^{2}} \cos ^{-1} x+x\right]+c \end{aligned}

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