1. anonymous

Set u=arctan(x), so that du= 1/(1+x^2). Thus, using this u-substitution, the integral is equivalent to the integral of sin(u). Since the integral of sin(u) is -cos(u)+C, the final answer is just $\sin (\tan^{-1} (x))+C$ Note also that sin(arctan(x)) is the same as $x/\sqrt{x^2+1}$ So that function plus your constant of integration is also a correct solution.

2. anonymous

Sorry, it should be $-\cos (\tan^{-1} (x))+C$

3. anonymous

thanx

4. anonymous

Also, that means that the alternate answer is instead $1/\sqrt{x^2+1} + C$