Solve ∫ (from 0 to 3) of (-39.5-7x)(x) wrt x

$$\int_{ 0 }^{ 3 } (-39.5-7x)(x) d x$$

Evaluate

$$-240.75$$

Solution Steps

Use the distributive property to multiply -39.5-7x by x.

$$\int _{0}^{3}-39.5x-7x^{2}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int -\frac{79x}{2}-7x^{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int -\frac{79x}{2}\mathrm{d}x+\int -7x^{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$-\frac{79\int x\mathrm{d}x}{2}-7\int x^{2}\mathrm{d}x$$

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 -39.5 times \frac{x^{2}}{2}.

$$-\frac{79x^{2}}{4}-7\int x^{2}\mathrm{d}x$$

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 -7 times \frac{x^{3}}{3}.

$$-\frac{79x^{2}}{4}-\frac{7x^{3}}{3}$$

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.

$$-\frac{79}{4}\times 3^{2}-\frac{7}{3}\times 3^{3}-\left(-\frac{79}{4}\times 0^{2}-\frac{7}{3}\times 0^{3}\right)$$

Simplify.

$$-\frac{963}{4}$$