Solve ∫ (from 0 to 4) of (-74+13.25x)(-0.125) wrt x

$$\int_{ 0 }^{ 4 } (-74+13.25x)(-0.125) d x$$

Evaluate

$$23.75$$

Solution Steps

Use the distributive property to multiply -74+13.25x by -0.125.

$$\int _{0}^{4}9.25-1.65625x\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 9.25-\frac{53x}{32}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 9.25\mathrm{d}x+\int -\frac{53x}{32}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 9.25\mathrm{d}x-\frac{53\int x\mathrm{d}x}{32}$$

Find the integral of 9.25 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{37x}{4}-\frac{53\int x\mathrm{d}x}{32}$$

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 -1.65625 times \frac{x^{2}}{2}.

$$\frac{37x}{4}-\frac{53x^{2}}{64}$$

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.

$$9.25\times 4-\frac{53}{64}\times 4^{2}-\left(9.25\times 0-\frac{53}{64}\times 0^{2}\right)$$

Simplify.

$$\frac{95}{4}$$