Solve ∫ (from 0 to 8) of (-16-8x+80/12*4+x)(1) wrt x

$$\int_{ 0 }^{ 8 } (-16-8x+ \frac{ 80 }{ 12 } \times 4+x)(1) d x$$

Evaluate

$$-\frac{416}{3}\approx -138.666666667$$

Solution Steps

Reduce the fraction \frac{80}{12} to lowest terms by extracting and canceling out 4.

$$\int _{0}^{8}\left(-16-8x+\frac{20}{3}\times 4+x\right)\times 1\mathrm{d}x$$

Express \frac{20}{3}\times 4 as a single fraction.

$$\int _{0}^{8}\left(-16-8x+\frac{20\times 4}{3}+x\right)\times 1\mathrm{d}x$$

Multiply 20 and 4 to get 80.

$$\int _{0}^{8}\left(-16-8x+\frac{80}{3}+x\right)\times 1\mathrm{d}x$$

Convert -16 to fraction -\frac{48}{3}.

$$\int _{0}^{8}\left(-\frac{48}{3}-8x+\frac{80}{3}+x\right)\times 1\mathrm{d}x$$

Since -\frac{48}{3} and \frac{80}{3} have the same denominator, add them by adding their numerators.

$$\int _{0}^{8}\left(\frac{-48+80}{3}-8x+x\right)\times 1\mathrm{d}x$$

Add -48 and 80 to get 32.

$$\int _{0}^{8}\left(\frac{32}{3}-8x+x\right)\times 1\mathrm{d}x$$

Combine -8x and x to get -7x.

$$\int _{0}^{8}\left(\frac{32}{3}-7x\right)\times 1\mathrm{d}x$$

Use the distributive property to multiply \frac{32}{3}-7x by 1.

$$\int _{0}^{8}\frac{32}{3}-7x\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int \frac{32}{3}-7x\mathrm{d}x$$

Integrate the sum term by term.

$$\int \frac{32}{3}\mathrm{d}x+\int -7x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int \frac{32}{3}\mathrm{d}x-7\int x\mathrm{d}x$$

Find the integral of \frac{32}{3} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{32x}{3}-7\int x\mathrm{d}x$$

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 -7 times \frac{x^{2}}{2}.

$$\frac{32x}{3}-\frac{7x^{2}}{2}$$

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.

$$\frac{32}{3}\times 8-\frac{7}{2}\times 8^{2}-\left(\frac{32}{3}\times 0-\frac{7}{2}\times 0^{2}\right)$$

Simplify.

$$-\frac{416}{3}$$