Solve ∫ (from 0 to 4) of (-100+29x)(-3) wrt x

$$\int_{ 0 }^{ 4 } (-100+29x)(-3) d x$$

Evaluate

$$504$$

Solution Steps

Use the distributive property to multiply -100+29x by -3.

$$\int _{0}^{4}300-87x\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 300-87x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 300\mathrm{d}x+\int -87x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 300\mathrm{d}x-87\int x\mathrm{d}x$$

Find the integral of 300 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$300x-87\int x\mathrm{d}x$$

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 -87 times \frac{x^{2}}{2}.

$$300x-\frac{87x^{2}}{2}$$

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.

$$300\times 4-\frac{87}{2}\times 4^{2}-\left(300\times 0-\frac{87}{2}\times 0^{2}\right)$$

Simplify.

$$504$$