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Provide Solution For  R.D.Sharma Maths Class 12 Chapter 18  Indefinite Integrals Exercise 18.18 Question 14 Maths Textbook Solution.

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Answer: \log \left|\sin ^{-1} x+\sqrt{9+\left(\sin ^{-1} x\right)^{2}}\right|+c

Hint Let \sin ^{-1} x=t

Given: \int \frac{1}{\sqrt{\left(1-x^{2}\right)\left(9+\left(\sin ^{-1} x\right)^{2}\right)}} d x


            \int \frac{1}{\sqrt{\left(1-x^{2}\right)\left(9+\left(\sin ^{-1} x\right)^{2}\right)}} d x            .........(1)

       Let \sin ^{-1} x=t

       \frac{1}{\sqrt{1-x^{2}} }dx=d t

From (1) we have

            \int \frac{d t}{\sqrt{9+t^{2}}}

           =\log \left|t+\sqrt{9+t^{2}}\right|+c

           =\log \left|\sin ^{-1} x+\sqrt{9+\left(\sin ^{-1} x\right)^{2}}\right|+c


       From (1) we have

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