#### Explain Solution R.D. Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18.21 Question 2 maths Textbook Solution.

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

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

Hint: Simplify it and solve it

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

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

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

Let

$x^{2}+2x-1=y$

$dx\left ( 2x+2 \right )=dy$

$I=\int \frac{d y}{\sqrt{y}}-\int \frac{1}{\sqrt{(x+1)^{2}-(\sqrt{2})^{2}}} d x$

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

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