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

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

Answers (1)

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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