Since $\int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1}$ for $k\neq -1$, replace $\int x^{2}\mathrm{d}x$ with $\frac{x^{3}}{3}$. Multiply $3$ times $\frac{x^{3}}{3}$.
$$x^{3}-2\int x\mathrm{d}x+\int c\mathrm{d}x$$
Since $\int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1}$ for $k\neq -1$, replace $\int x\mathrm{d}x$ with $\frac{x^{2}}{2}$. Multiply $-2$ times $\frac{x^{2}}{2}$.
$$x^{3}-x^{2}+\int c\mathrm{d}x$$
Find the integral of $c$ using the table of common integrals rule $\int a\mathrm{d}x=ax$.
$$x^{3}-x^{2}+cx$$
The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.
