Write a value of `intsqrt(9+x^2)\ dx`

Explanation

Solution:

`I=\int \sqrt{9+x^{2}} d x`

We know that, `\int \sqrt{x^{2}+a^{2}} d x=\frac{x}{2} \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \log |x+\sqrt{x^{2}+a^{2}} |+c`

Replace `\{x} \Rightarrow \{x} and \{a} \Rightarrow 3` we get

`\int \sqrt{3^{2}+x^{2}} d x &=\frac{x}{2} \sqrt{3^{2}-x^{2}}+\frac{3^{2}}{2} \log |x+\sqrt{x^{2}+3^{2}} |+c`

=`\frac{x}{2} \sqrt{9-x^{2}}+\frac{9}{2} \log |x+\sqrt{x^{2}+9}|+c`