Solve ∫ (from 0 to 2) of x(3+2x) wrt x

$$\int _ { 0 } ^ { 2 } x ( 3 + 2 x ) d x$$

Evaluate

$$\frac{34}{3}\approx 11.333333333$$

Solution Steps

Use the distributive property to multiply x by 3+2x.

$$\int _{0}^{2}3x+2x^{2}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 3x+2x^{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 3x\mathrm{d}x+\int 2x^{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$3\int x\mathrm{d}x+2\int x^{2}\mathrm{d}x$$

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 3 times \frac{x^{2}}{2}.

$$\frac{3x^{2}}{2}+2\int x^{2}\mathrm{d}x$$

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 2 times \frac{x^{3}}{3}.

$$\frac{3x^{2}}{2}+\frac{2x^{3}}{3}$$

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.

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

Simplify.

$$\frac{34}{3}$$