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Provide Solution For  R.D. Sharma Maths Class 12 Chapter 18  Indefinite Integrals Exercise 18.21 Question 7 Maths Textbook Solution.

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

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

Hint: Simplify the given function


        \begin{aligned} &I=\int \frac{x+2}{\sqrt{x^{2}+2 x-1}} d x \\ &I=\frac{1}{2} \int \frac{2 x+4}{\sqrt{x^{2}+2 x-1}} d x \\ &I=\frac{1}{2} \int \frac{2 x+2}{\sqrt{x^{2}+2 x-1}} d x+\frac{2}{2} \int \frac{1}{\sqrt{x^{2}+2 x-1}} d x \\ &I=\frac{1}{2} \int \frac{2 x+2}{\sqrt{x^{2}+2 x-1}} d x+1 \int \frac{1}{\sqrt{x^{2}+2 x+1-2}} d x \\ &I=\frac{1}{2} \int \frac{2 x+2}{\sqrt{x^{2}+2 x-1}} d x+\int \frac{1}{\sqrt{(x+1)^{2}-(\sqrt{2})^{2}}} d x \end{aligned}

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

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

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