Question

Write a value of $\int \sqrt{9+x^2}dx$.

Answer 1

$$\begin{gathered} \int \sqrt{9+x^2}dx \\ =\int \sqrt{3^2+x^2}dx\left(\because \sqrt{a^2+x^2}=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\ln \left|x+\sqrt{x^2+a^2}\right|\right) \\ =\frac{x}{2}\sqrt{9+x^2}+\frac{9}{2}\ln \left|x+\sqrt{9+x^2}\right|+C \end{gathered}$$