Solve ∫ (from 0 to 4) of (-43+8.375x)(0.5) wrt x

$$\int_{ 0 }^{ 4 } (-43+8.375x)(0.5) d x$$

Evaluate

$$-52.5$$

Solution Steps

Use the distributive property to multiply -43+8.375x by 0.5.

$$\int _{0}^{4}-21.5+4.1875x\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int -21.5+\frac{67x}{16}\mathrm{d}x$$

Integrate the sum term by term.

$$\int -21.5\mathrm{d}x+\int \frac{67x}{16}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int -21.5\mathrm{d}x+\frac{67\int x\mathrm{d}x}{16}$$

Find the integral of -21.5 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$-\frac{43x}{2}+\frac{67\int x\mathrm{d}x}{16}$$

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 4.1875 times \frac{x^{2}}{2}.

$$-\frac{43x}{2}+\frac{67x^{2}}{32}$$

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.

$$-21.5\times 4+\frac{67}{32}\times 4^{2}-\left(-21.5\times 0+\frac{67}{32}\times 0^{2}\right)$$

Simplify.

$$-\frac{105}{2}$$