Solve ∫ (from 0 to 4) of (-87.5+38.5x)(x) wrt x

$$\int_{ 0 }^{ 4 } (-87.5+38.5x)(x) d x$$

Evaluate

$$\frac{364}{3}\approx 121.333333333$$

Solution Steps

Use the distributive property to multiply -87.5+38.5x by x.

$$\int _{0}^{4}-87.5x+38.5x^{2}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int \frac{-175x+77x^{2}}{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int -\frac{175x}{2}\mathrm{d}x+\int \frac{77x^{2}}{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\frac{-175\int x\mathrm{d}x+77\int x^{2}\mathrm{d}x}{2}$$

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 -87.5 times \frac{x^{2}}{2}.

$$-\frac{175x^{2}}{4}+\frac{77\int x^{2}\mathrm{d}x}{2}$$

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

$$-\frac{175x^{2}}{4}+\frac{77x^{3}}{6}$$

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.

$$-\frac{175}{4}\times 4^{2}+\frac{77}{6}\times 4^{3}-\left(-\frac{175}{4}\times 0^{2}+\frac{77}{6}\times 0^{3}\right)$$

Simplify.

$$\frac{364}{3}$$