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Please Solve RD Sharma Class 12 Chapter 18 Indefinite Integrals Exercise18.32 Question 12 Maths Textbook Solution.

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

Hint :-                Use substitution method to solve this integral.

Given :-              \int \frac{1}{(x-1) \sqrt{x+2}} d x

Sol : -               Let      I=\int \frac{1}{(x-1) \sqrt{x+2}} d x

          Put         x+2=t^{2}\Rightarrow dx=2tdt

                      \begin{aligned} &I=\int \frac{1}{\left(t^{2}-2-1\right) \sqrt{t^{2}}} \quad 2 t d t \quad\left(\because x=t^{2}-2\right) \\ &=2 \int \frac{1}{\left(t^{2}-3\right) t} t d t \\ &=2 \int \frac{1}{t^{2}-3} d t \end{aligned}

                     \begin{array}{ll} =2 \cdot \frac{1}{2 \sqrt{3}} \log \left|\frac{t-\sqrt{3}}{t+\sqrt{3}}\right|+C & \left(\because \int \frac{1}{x^{2}-a^{2}} d x=1 / 2 a \log \left|\frac{x-a}{x+a}\right|+C\right) \\\\ =\frac{1}{\sqrt{3}} \log \left|\frac{\sqrt{x+2}-\sqrt{3}}{\sqrt{x+2}+\sqrt{3}}\right|+C \quad & (\because t=\sqrt{x+2}) \text { Ans.. } \end{array}     


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