Solve ∫ (from 2 to 4) of 6t^2-12tdt

$$\int _ { 2 } ^ { 4 } 6 t ^ { 2 } - 12 t d t$$

Evaluate

$$40$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 6t^{2}-12t\mathrm{d}t$$

Integrate the sum term by term.

$$\int 6t^{2}\mathrm{d}t+\int -12t\mathrm{d}t$$

Factor out the constant in each of the terms.

$$6\int t^{2}\mathrm{d}t-12\int t\mathrm{d}t$$

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

$$2t^{3}-12\int t\mathrm{d}t$$

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 -12 times \frac{t^{2}}{2}.

$$2t^{3}-6t^{2}$$

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.

$$2\times 4^{3}-6\times 4^{2}-\left(2\times 2^{3}-6\times 2^{2}\right)$$

Simplify.

$$40$$