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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise 18.14 Question 6

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

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

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


Let  I=\int \frac{1}{\sqrt{a^{2}+b^{2} x^{2}}} d x

\text { Put } b x=t \Rightarrow b d x=d t \Rightarrow d x=\frac{d t}{b} \text { then }

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

=\frac{1}{b} \log \left|b x+\sqrt{a^{2}+b^{2} x^{2}}\right|+c \quad\quad\quad\quad\quad\quad\quad[\because t=b x]



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