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Differentiate w.r.t. x
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\int t^{2}-t\mathrm{d}t
Evaluate the indefinite integral first.
\int t^{2}\mathrm{d}t+\int -t\mathrm{d}t
Integrate the sum term by term.
\int t^{2}\mathrm{d}t-\int t\mathrm{d}t
Factor out the constant in each of the terms.
\frac{t^{3}}{3}-\int t\mathrm{d}t
Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{2}\mathrm{d}t with \frac{t^{3}}{3}.
\frac{t^{3}}{3}-\frac{t^{2}}{2}
Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t\mathrm{d}t with \frac{t^{2}}{2}. Multiply -1 times \frac{t^{2}}{2}.
\frac{x^{3}}{3}-\frac{x^{2}}{2}-\left(\frac{0^{3}}{3}-\frac{0^{2}}{2}\right)
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.
-\frac{x^{2}}{2}+\frac{x^{3}}{3}
Simplify.