Solve ∫ (from -1 to 3) of -x^3+8x^2+2x-13div5 wrt x

$$\int_{ -1 }^{ 3 } - { x }^{ 3 } +8 { x }^{ 2 } +2x-13 \div 5 d x$$

Evaluate

$$\frac{784}{15}\approx 52.266666667$$

Solution Steps

Evaluate the indefinite integral first.

$$\int -x^{3}+8x^{2}+2x-\frac{13}{5}\mathrm{d}x$$

Integrate the sum term by term.

$$\int -x^{3}\mathrm{d}x+\int 8x^{2}\mathrm{d}x+\int 2x\mathrm{d}x+\int -\frac{13}{5}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$-\int x^{3}\mathrm{d}x+8\int x^{2}\mathrm{d}x+2\int x\mathrm{d}x+\int -\frac{13}{5}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}. Multiply -1 times \frac{x^{4}}{4}.

$$-\frac{x^{4}}{4}+8\int x^{2}\mathrm{d}x+2\int x\mathrm{d}x+\int -\frac{13}{5}\mathrm{d}x$$

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

$$-\frac{x^{4}}{4}+\frac{8x^{3}}{3}+2\int x\mathrm{d}x+\int -\frac{13}{5}\mathrm{d}x$$

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

$$-\frac{x^{4}}{4}+\frac{8x^{3}}{3}+x^{2}+\int -\frac{13}{5}\mathrm{d}x$$

Find the integral of -\frac{13}{5} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$-\frac{x^{4}}{4}+\frac{8x^{3}}{3}+x^{2}-\frac{13x}{5}$$

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.

$$-\frac{3^{4}}{4}+\frac{8}{3}\times 3^{3}+3^{2}-\frac{13}{5}\times 3-\left(-\frac{\left(-1\right)^{4}}{4}+\frac{8}{3}\left(-1\right)^{3}+\left(-1\right)^{2}-\frac{13}{5}\left(-1\right)\right)$$

Simplify.

$$\frac{784}{15}$$