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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 36

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Answer: \frac{1}{2} x \sqrt{9+x^{2}}+\frac{9}{2} \log \left|x+\sqrt{9+x^{2}}\right|+c

Hint: You must know about the integral rule of trigonometric functions

Given: \int \sqrt{9+x^{2}}dx


\int \sqrt{9+x^{2}}dx

We know that \int \sqrt{x^{2}+a^{2}} d x=\frac{x}{2} \sqrt{a^{2}+x^{2}}+\frac{a^{2}}{2} \log \left|x+\sqrt{x^{2}+a^{2}}\right|+c

Replace x ->x and a->3

\begin{aligned} &\int \sqrt{3^{2}+x^{2}} d x=\frac{x}{2} \sqrt{3^{2}+x^{2}}+\frac{3^{2}}{2} \log \left|x+\sqrt{x^{2}+3^{2}}\right|+c \\ &=\frac{1}{2} x \sqrt{9+x^{2}}+\frac{9}{2} \log \left|x+\sqrt{x^{2}+9}\right|+c \end{aligned}

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