Solve ∫ (from 1 to 100) of x^2+9

$$\int _ { 1 } ^ { 100 } x ^ { 2 } + 9$$

Evaluate

$$334224$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{2}+9\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{2}\mathrm{d}x+\int 9\mathrm{d}x$$

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}.

$$\frac{x^{3}}{3}+\int 9\mathrm{d}x$$

Find the integral of 9 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{3}}{3}+9x$$

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.

$$\frac{100^{3}}{3}+9\times 100-\left(\frac{1^{3}}{3}+9\times 1\right)$$

Simplify.

$$334224$$