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Please Solve R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18 Point 18 Question 3 Maths Textbook Solution.

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Answer:\sin ^{-1}\left(\frac{e^{x}}{4}\right)+c

Hint Let e^{x}=t

Given: \int \frac{e^{x}}{\sqrt{16-e^{2 x}}} d x


            \int \frac{e^{x}}{\sqrt{16-e^{2 x}}} d x                                ..........(1)

         Let e^{x}=t

         e^{x} d x=d t

         From (1) we have

        \int \frac{d t}{\sqrt{(4)^{2}-t^{2}}}                                        \left[\because \int \frac{d t}{\sqrt{a^{2}-x^{2}}}=\sin ^{-1} \frac{x}{a}\right]

        =\sin ^{-1}\left(\frac{t}{4}\right)+c

        =\sin ^{-1}\left(\frac{e^{x}}{4}\right)+c

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