Solve ∫ (from 0 to 8) of 2.5x^3+1.25x^2 wrt x

$$\int_{ 0 }^{ 8 } 2.5 { x }^{ 3 } +1.25 { x }^{ 2 } d x$$

Evaluate

$$\frac{8320}{3}\approx 2773.333333333$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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 2.5 times \frac{x^{4}}{4}.

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

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

$$\frac{5x^{4}}{8}+\frac{5x^{3}}{12}$$

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.

$$\frac{5}{8}\times 8^{4}+\frac{5}{12}\times 8^{3}-\left(\frac{5}{8}\times 0^{4}+\frac{5}{12}\times 0^{3}\right)$$

Simplify.

$$\frac{8320}{3}$$