Solve ∫ (from t to t + 1) of (-x^2+2x) wrt x

$$\int _ { t } ^ { t + 1 } ( - x ^ { 2 } + 2 x ) d x$$

Evaluate

$$\frac{2}{3}+t-t^{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int -x^{2}+2x\mathrm{d}x$$

Integrate the sum term by term.

$$\int -x^{2}\mathrm{d}x+\int 2x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$-\int x^{2}\mathrm{d}x+2\int x\mathrm{d}x$$

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 -1 times \frac{x^{3}}{3}.

$$-\frac{x^{3}}{3}+2\int x\mathrm{d}x$$

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}.

$$-\frac{x^{3}}{3}+x^{2}$$

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.

$$-\frac{1}{3}\left(t+1\right)^{3}+\left(t+1\right)^{2}-\left(-\frac{t^{3}}{3}+t^{2}\right)$$

Simplify.

$$\frac{2}{3}+t-t^{2}$$

Differentiate w.r.t. t

$$1-2t$$