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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise 18.14 Question 2

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Answer:  \frac{1}{2 a b} \log \left|\frac{a x-b}{a x+b}\right|+c

Hint: To solve this integral, use special integral formula.

Given: \int \frac{1}{a^{2} x^{2}-b^{2}} d x

Solution: Let  I=\int \frac{1}{a^{2} x^{2}-b^{2}} d x \: x=\frac{1}{a^{2}} \int \frac{1}{x^{2}-\frac{b^{2}}{a^{2}}} d x

=\frac{1}{a^{2}} \int \frac{1}{x^{2}-\left(\frac{b}{a}\right)^{2}} d x

=\frac{1}{a^{2}} \cdot \frac{1}{2 \times \frac{b}{a}} \log \left|\frac{x-\frac{b}{a}}{x+\frac{b}{a}}\right|+c \quad \quad \quad \quad \quad \quad \quad\left[\because \int \frac{1}{x^{2}-a^{2}} d x=\frac{1}{2 a} \log \left|\frac{x-a}{x+a}\right|+c\right]

\begin{aligned} &=\frac{1}{2 a b} \log \left|\frac{\frac{a x-b}{a}}{\frac{a x+b}{a}}\right|+c \\ \end{aligned}

=\frac{1}{2 a b} \log \left|\frac{a x-b}{a x+b}\right|+c


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