Question

How do you integrate `intxsqrt(x^2 + 1) dx`?

Answer 1

The answer is `\frac{1}{3}(x^{2}+1)^{3/2}+C=\frac{1}{3}(x^{2}+1)\sqrt{x^{2}+1}+C`.

You can either find this answer by doing some educated guessing or by doing a substitution. You can let `u=x^{2}+1` so that `du=2x dx` and your integral becomes `\int\frac{1}{2}u^{1/2}du=\frac{1}{2}\frac{u^{3/2}}{3/2}+C=\frac{1}{2}\cdot\frac{2}{3}u^{3/2}+C=\frac{1}{3}u^{3/2}+C`. Now plug `u=x^{2}+1` to get the same answer as above:

`\int x\sqrt{x^{2}+1}dx=\frac{1}{3}(x^{2}+1)^{3/2}+C=\frac{1}{3}(x^{2}+1)\sqrt{x^{2}+1}+C`