Solve ∫ (from 1 to 3) of left(2x+1right)left(3-xright) wrt x

$$\int_{ 1 }^{ 3 } \left( 2x+1 \right) \left( 3-x \right) d x$$

Evaluate

$$\frac{26}{3}\approx 8.666666667$$

Solution Steps

Apply the distributive property by multiplying each term of 2x+1 by each term of 3-x.

$$\int _{1}^{3}6x-2x^{2}+3-x\mathrm{d}x$$

Combine 6x and -x to get 5x.

$$\int _{1}^{3}5x-2x^{2}+3\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 5x-2x^{2}+3\mathrm{d}x$$

Integrate the sum term by term.

$$\int 5x\mathrm{d}x+\int -2x^{2}\mathrm{d}x+\int 3\mathrm{d}x$$

Factor out the constant in each of the terms.

$$5\int x\mathrm{d}x-2\int x^{2}\mathrm{d}x+\int 3\mathrm{d}x$$

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

$$\frac{5x^{2}}{2}-2\int x^{2}\mathrm{d}x+\int 3\mathrm{d}x$$

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}.

$$\frac{5x^{2}}{2}-\frac{2x^{3}}{3}+\int 3\mathrm{d}x$$

Find the integral of 3 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{5x^{2}}{2}-\frac{2x^{3}}{3}+3x$$

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.

$$\frac{5}{2}\times 3^{2}-\frac{2}{3}\times 3^{3}+3\times 3-\left(\frac{5}{2}\times 1^{2}-\frac{2}{3}\times 1^{3}+3\times 1\right)$$

Simplify.

$$\frac{26}{3}$$