Solve ∫ (from 1 to 2) of (3t^2-t)dt

$$\int _ { 1 } ^ { 2 } ( 3 t ^ { 2 } - t ) d t$$

Evaluate

$$\frac{11}{2}=5.5$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 3t^{2}-t\mathrm{d}t$$

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

$$3\int t^{2}\mathrm{d}t-\int t\mathrm{d}t$$

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}. Multiply 3 times \frac{t^{3}}{3}.

$$t^{3}-\int t\mathrm{d}t$$

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

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

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.

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

Simplify.

$$\frac{11}{2}$$