Answer
$\frac{x^{2}}{2}+x+C$
Work Step by Step
$\int(x+1)dx= \int xdx +\int1dx$
$\int xdx = \frac{x^{2}}{2}+ C_{1}$ (As $\int x^{n}dx= \frac{x^{n+1}}{n+1}+ C_{1})$.
$\int1dx = x+C_{2}$ (As $\frac{d}{dx}(x+C_{2})=1$).
Now, $\int(x+1)dx= \frac{x^{2}}{2}+x+(C_{1}+C_{2}) $= $\frac{x^{2}}{2}+x+C$ (Note that $C_{1}$ and $C_{2}$ are arbitrary constants and C is the sum of these two constants which is again a constant).
$\frac{d}{dx}(\frac{x^{2}}{2}+x+C)= \frac{2x}{2}+1+0$= x+1.
Here derivative of the integral is the integrand. Hence the answer is correct.
