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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise 18.14 Question 7

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Answer: \frac{1}{b} \sin ^{-1}\left(\frac{b x}{a}\right)+c

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

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


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

\begin{aligned} &\text { Put } b x=t \Rightarrow b d x=d t \Rightarrow d x=\frac{d t}{b} \text { then } \\ &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 \end{aligned}

\begin{aligned} &=\frac{1}{b} \sin ^{-1}\left(\frac{t}{a}\right)+c \quad\left[\because \int \frac{1}{\sqrt{a^{2}-x^{2}}} d x=\sin ^{-1}\left(\frac{x}{a}\right)+c\right] \\\\ &=\frac{1}{b} \sin ^{-1}\left(\frac{b x}{a}\right)+c \quad[\because t=b x] \end{aligned}

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