Solve ∫ (from 3 to 50) of x(x+8) wrt x

$$\int _ { 3 } ^ { 50 } x ( x + 8 ) d x$$

Evaluate

$$\frac{154865}{3}\approx 51621.666666667$$

Solution Steps

Use the distributive property to multiply x by x+8.

$$\int _{3}^{50}x^{2}+8x\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int x^{2}+8x\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{2}\mathrm{d}x+\int 8x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x^{2}\mathrm{d}x+8\int x\mathrm{d}x$$

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

$$\frac{x^{3}}{3}+8\int x\mathrm{d}x$$

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

$$\frac{x^{3}}{3}+4x^{2}$$

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.

$$\frac{50^{3}}{3}+4\times 50^{2}-\left(\frac{3^{3}}{3}+4\times 3^{2}\right)$$

Simplify.

$$\frac{154865}{3}$$