Solve ∫ (from 0 to 2) of (60-1/2t)dt

$$\int _ { 0 } ^ { 2 } ( 60 - \frac { 1 } { 2 } t ) d t$$

Evaluate

$$119$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 60-\frac{t}{2}\mathrm{d}t$$

Integrate the sum term by term.

$$\int 60\mathrm{d}t+\int -\frac{t}{2}\mathrm{d}t$$

Factor out the constant in each of the terms.

$$\int 60\mathrm{d}t-\frac{\int t\mathrm{d}t}{2}$$

Find the integral of 60 using the table of common integrals rule \int a\mathrm{d}t=at.

$$60t-\frac{\int t\mathrm{d}t}{2}$$

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 -\frac{1}{2} times \frac{t^{2}}{2}.

$$60t-\frac{t^{2}}{4}$$

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.

$$60\times 2-\frac{2^{2}}{4}-\left(60\times 0-\frac{0^{2}}{4}\right)$$

Simplify.

$$119$$