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provide solution for RD Sharma maths class 12 chapter Indefinite Integrals exercise  18.9 question 34

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Answer: \frac{3}{2}\left(x^{2}-1\right)^{\frac{2}{3}}+c

Hint:Use substitution method to solve this integral.

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


        \begin{aligned} &\text { Let } I=\int \frac{2 x}{\sqrt[3]{x^{2}-1}} d x \\ &\text { Put } x^{2}-1=t \Rightarrow 2 x d x=d t \text { then } \end{aligned}

        \Rightarrow I=\int \frac{1}{\sqrt[3]{t}} d t=\int \frac{1}{t^{\frac{1}{3}}} d t=\int t^{-\frac{1}{3}} d t

                  =\frac{t^{-\frac{1}{3}+1}}{-\frac{1}{3}+1}+c=\frac{t^{\frac{2}{3}}}{\frac{2}{3}}+c                                \left[\because \int x^{n} d x=\frac{x^{n+1}}{n+1}+c\right]

                  \begin{aligned} &=\frac{3}{2} t^{\frac{2}{3}} \\ &=\frac{3}{2}\left(x^{2}-1\right)^{\frac{2}{3}}+c \quad\left[\because t=x^{2}-1\right] \end{aligned}


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