Solve ∫ (from 0 to 4) of (12x+6x^2) wrt x

$$\int_{ 0 }^{ 4 } (12x+6 { x }^{ 2 } ) d x$$

Evaluate

$$224$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply 12 times \frac{x^{2}}{2}.

$$6x^{2}+6\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply 6 times \frac{x^{3}}{3}.

$$6x^{2}+2x^{3}$$

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.

$$6\times 4^{2}+2\times 4^{3}-\left(6\times 0^{2}+2\times 0^{3}\right)$$

Simplify.

$$224$$