Solve ∫ (from 0 to 4) of 147000x(6-x) wrt x

$$\int_{ 0 }^{ 4 } 147000x(6-x) d x$$

Evaluate

$$3920000$$

Solution Steps

Use the distributive property to multiply 147000x by 6-x.

$$\int _{0}^{4}882000x-147000x^{2}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 882000x-147000x^{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 882000x\mathrm{d}x+\int -147000x^{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$882000\int x\mathrm{d}x-147000\int x^{2}\mathrm{d}x$$

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 882000 times \frac{x^{2}}{2}.

$$441000x^{2}-147000\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply -147000 times \frac{x^{3}}{3}.

$$441000x^{2}-49000x^{3}$$

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.

$$441000\times 4^{2}-49000\times 4^{3}-\left(441000\times 0^{2}-49000\times 0^{3}\right)$$

Simplify.

$$3920000$$