Solve ∫ (from 0 to 4) of (-8-4.5+35x)(-3) wrt x

$$\int_{ 0 }^{ 4 } (-8-4.5+35x)(-3) d x$$

Evaluate

$$-690$$

Solution Steps

Subtract 4.5 from -8 to get -12.5.

$$\int _{0}^{4}\left(-12.5+35x\right)\left(-3\right)\mathrm{d}x$$

Use the distributive property to multiply -12.5+35x by -3.

$$\int _{0}^{4}37.5-105x\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 37.5-105x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 37.5\mathrm{d}x+\int -105x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 37.5\mathrm{d}x-105\int x\mathrm{d}x$$

Find the integral of 37.5 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{75x}{2}-105\int x\mathrm{d}x$$

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 -105 times \frac{x^{2}}{2}.

$$\frac{75x-105x^{2}}{2}$$

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.

$$37.5\times 4-\frac{105}{2}\times 4^{2}-\left(37.5\times 0-\frac{105}{2}\times 0^{2}\right)$$

Simplify.

$$-690$$