Solve ∫ (from 0 to 2) of 2+12t^2dt

$$\int _ { 0 } ^ { 2 } 2 + 12 t ^ { 2 } d t$$

Evaluate

$$36$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

Find the integral of 2 using the table of common integrals rule \int a\mathrm{d}t=at.

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

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

$$2t+4t^{3}$$

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.

$$2\times 2+4\times 2^{3}-\left(2\times 0+4\times 0^{3}\right)$$

Simplify.

$$36$$