What is the antiderivative of x(x+1)?

Jun 21, 2016

$F \left(x\right) = {x}^{3} / 3 + {x}^{2} / 2 + C$

Explanation:

Before attempting to integrate, the expression should be simplified by distributing $x$ over $\left(x + 1\right)$.

Thus,

$f \left(x\right) = x \left(x + 1\right)$
$f \left(x\right) = \left(x \cdot x\right) + \left(x \cdot 1\right)$
$f \left(x\right) = {x}^{2} + x$

Integrating $f \left(x\right)$, we obtain the following:

$F \left(x\right) = \int f \left(x\right) \mathrm{dx}$
$F \left(x\right) = \int \left({x}^{2} + x\right) \mathrm{dx}$
$F \left(x\right) = {x}^{2 + 1} / \left(2 + 1\right) + {x}^{1 + 1} / \left(1 + 1\right) + C$
$F \left(x\right) = {x}^{3} / 3 + {x}^{2} / 2 + C$

Note: Finding the anti-derivative of a function is the same as integration.