Solve ∫ (from 0 to 3) of (-37.5+37x)(-3) wrt x

$$\int_{ 0 }^{ 3 } (-37.5+37x)(-3) d x$$

Evaluate

$$-162$$

Solution Steps

Use the distributive property to multiply -37.5+37x by -3.

$$\int _{0}^{3}112.5-111x\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 112.5-111x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 112.5\mathrm{d}x+\int -111x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 112.5\mathrm{d}x-111\int x\mathrm{d}x$$

Find the integral of 112.5 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{225x}{2}-111\int x\mathrm{d}x$$

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 -111 times \frac{x^{2}}{2}.

$$\frac{225x-111x^{2}}{2}$$

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.

$$112.5\times 3-\frac{111}{2}\times 3^{2}-\left(112.5\times 0-\frac{111}{2}\times 0^{2}\right)$$

Simplify.

$$-162$$