Solve ∫ (from 0 to t) of (t-s)ds

$$\int _ { 0 } ^ { t } ( t - s ) d s$$

Evaluate

$$\frac{t^{2}}{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int t-s\mathrm{d}s$$

Integrate the sum term by term.

$$\int t\mathrm{d}s+\int -s\mathrm{d}s$$

Factor out the constant in each of the terms.

$$\int t\mathrm{d}s-\int s\mathrm{d}s$$

Find the integral of t using the table of common integrals rule \int a\mathrm{d}s=as.

$$ts-\int s\mathrm{d}s$$

Since \int s^{k}\mathrm{d}s=\frac{s^{k+1}}{k+1} for k\neq -1, replace \int s\mathrm{d}s with \frac{s^{2}}{2}. Multiply -1 times \frac{s^{2}}{2}.

$$ts-\frac{s^{2}}{2}$$

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.

$$tt-\frac{t^{2}}{2}-\left(t\times 0-\frac{0^{2}}{2}\right)$$

Simplify.

$$\frac{t^{2}}{2}$$

Differentiate w.r.t. t

$$t$$