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Provide Solution For  R.D.Sharma Maths Class 12 Chapter 18  Indefinite Integrals Exercise 18.18 Question 13 Maths Textbook Solution.

Answers (1)

Answer: 3 \log \left|x^{\frac{1}{3}}+\sqrt{x^{\frac{2}{3}}-4}\right|+c

Hint Let x^{\frac{1}{3}}=t

Given: \int \frac{1}{x^{\frac{2}{3}} \sqrt{x^{\frac{2}{3}}-4}} d x


            \int \frac{1}{x^{\frac{2}{3}} \sqrt{x^{\frac{2}{3}}-4}} d x                        ............(1)

            Let  x^{\frac{1}{3}}=t

            \frac{1}{3} x^{\frac{-2}{3}} d x=d t

            \frac{d x}{x^{\frac{2}{3}}}=3 d t                                                    (Differentiate w.r.t to t)

Put in (1) we have

              3 \int \frac{d t}{\sqrt{t^{2}-4}}

            =3 \log \left|t+\sqrt{t^{2}-4}\right|+c

            =3 \log \left|x^{\frac{1}{3}}+\sqrt{x^{\frac{2}{3}}-4}\right|+c \quad\left[\because \int \frac{d x}{\sqrt{x^{2}-a^{2}}}=\log \left|x+\sqrt{x^{2}-a^{2}}\right|+c\right]


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