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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise 18.14 Question 1

Answers (1)

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

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

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

Solution: Let

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

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

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

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





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