Solve ∫ (from 0 to 4) of (-74+13.25x)(0.5) wrt x

$$\int_{ 0 }^{ 4 } (-74+13.25x)(0.5) d x$$

Evaluate

$$-95$$

Solution Steps

Use the distributive property to multiply -74+13.25x by 0.5.

$$\int _{0}^{4}-37+6.625x\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int -37+\frac{53x}{8}\mathrm{d}x$$

Integrate the sum term by term.

$$\int -37\mathrm{d}x+\int \frac{53x}{8}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int -37\mathrm{d}x+\frac{53\int x\mathrm{d}x}{8}$$

Find the integral of -37 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$-37x+\frac{53\int x\mathrm{d}x}{8}$$

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 6.625 times \frac{x^{2}}{2}.

$$-37x+\frac{53x^{2}}{16}$$

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.

$$-37\times 4+\frac{53}{16}\times 4^{2}-\left(-37\times 0+\frac{53}{16}\times 0^{2}\right)$$

Simplify.

$$-95$$