Question

`\int{\sqrt{9+x^(2) } } \, dx`

Answer 1

This is the integral

`(9)/(2)\left(sin^(-1)\left((x)/(3) \right)+(x)/(3)\sqrt{1-\left((x)/(3)^2 \right)} \right) + C`

Explanation:

Setup/solve

`sin^2A+cos^2A=1\\\mathrm{and}\:\: cos^2A=2cos^2A-1\\sin^2A=2sinA \cdot cosA\\\mathrm{where}\:\: x=3sin\:t\\\int√(9-x^2) dx=\int3√(1-sin^2t) \cdot 3cos\:tdt=9\int cos^2\:tdt\\9\int cos^2\:tdx = (9)/(2) \int(1 + cos2t)dt = (9)/(2)\left(t + (1)/(2)sin\:2t \right) + C`

Lastly, describe this in terms of x

`\int √(9-x^2)dx=(9)/(2)\left(sin^(-1)\left((x)/(3) \right)+(x)/(3)\sqrt{1-\left((x)/(3)^2 \right)} \right) + C`

I hope this helps, and have a good day!