#### Provide Solution For R.D.Sharma Maths Class 12 Chapter 18  Indefinite Integrals Exercise 18.19 Question 3 Maths Textbook Solution.

Answer: $-\frac{2}{\sqrt{5}} \log \left|\frac{x+1-\sqrt{5}}{x+1+\sqrt{5}}\right|+\frac{1}{2} \log \left|x^{2}+2 x-4\right|+c$

Hint Find value of A and B

Given: $\int \frac{x-3}{x^{2}+2 x-4} d x$

Solution: Let $x-3=A+B \frac{d}{d x}\left(x^{2}+2 x-4\right)$

$x-3=A+B(2 x+2)$

$x-3=A+2 B x+2 B$

On comparing,

$\left[\begin{array}{c} x=2 B x \Rightarrow B=\frac{1}{2} \\ -3=A+2 B \Rightarrow-3-2 \times \frac{1}{2}=A \Rightarrow A=-4 \end{array}\right]$

$=-4 \int \frac{d x}{x^{2}+2 x-4}+\frac{1}{2} \int \frac{\frac{d}{d x}\left(x^{2}+2 x-4\right)}{x^{2}+2 x-4}$                                                        $\left[\begin{array}{l} x^{2}+2 x+1-4-1 \\ (x+1)^{2}-(\sqrt{5})^{2} \end{array}\right]$

$=-4 \int \frac{d x}{(x+1)^{2}-(\sqrt{5})^{2}}+\frac{1}{2} \log \left|x^{2}+2 x-4\right|$                                                        $\left[\int \frac{d x}{x^{2}-a^{2}}=\frac{1}{2 a} \log \left|\frac{x-a}{x+a}\right|+c\right]$

$=-\frac{2}{\sqrt{5}} \log \left|\frac{x+1-\sqrt{5}}{x+1+\sqrt{5}}\right|+\frac{1}{2} \log \left|x^{2}+2 x-4\right|+c$