Solve ∫ (from 0 to 30) of x^2+5*4

$$\int _ { 0 } ^ { 30 } x ^ { 2 } + 5 \cdot 4$$

Evaluate

$$9600$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{2}+4\times 5\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{2}\mathrm{d}x+\int 4\times 5\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x^{2}\mathrm{d}x+4\int 5\mathrm{d}x$$

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}.

$$\frac{x^{3}}{3}+4\int 5\mathrm{d}x$$

Find the integral of 5 using the table of common integrals rule \int a\mathrm{d}x=ax. Multiply 4 times 5x.

$$\frac{x^{3}}{3}+20x$$

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.

$$\frac{30^{3}}{3}+20\times 30-\left(\frac{0^{3}}{3}+20\times 0\right)$$

Simplify.

$$9600$$