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Please Solve R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18.21 Question 8 Maths Textbook Solution.

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

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

Hint: Simplify the given function

Solution: I=\int \frac{x+2}{\sqrt{x^{2}-1}} d x

           \begin{aligned} &I=\frac{1}{2} \int \frac{2 x}{\sqrt{x^{2}-1}} d x+2 \int \frac{1}{\sqrt{x^{2}-1}} d x \\ &I=\frac{1}{2}\left[\frac{x^{2}-1}{\frac{1}{2}}\right]+2 \log \left|x+\sqrt{x^{2}-1}\right|+c \end{aligned}

           \left[\begin{array}{l} U \sin g \\ \int(f(x))^{n} f^{1}(x) d x=\frac{f(x)^{n+1}}{n+1}+c \\ \int \frac{1}{\sqrt{x^{2}-a^{2}}} d x=\log \left|x+\sqrt{x^{2}-a^{2}}\right|+c \end{array}\right]

           I=\sqrt{x^{2}-1}+2 \log \left|x+\sqrt{x^{2}-1}\right|+c


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