Solve ∫ (from 0 to 2) of x^2+5+x^2

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

Evaluate

$$\frac{46}{3}\approx 15.333333333$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

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}.

$$\frac{x^{3}}{3}+\int 5\mathrm{d}x+\int x^{2}\mathrm{d}x$$

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

$$\frac{x^{3}}{3}+5x+\int x^{2}\mathrm{d}x$$

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}.

$$\frac{x^{3}}{3}+5x+\frac{x^{3}}{3}$$

Simplify.

$$\frac{2x^{3}}{3}+5x$$

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.

$$\frac{2}{3}\times 2^{3}+5\times 2-\left(\frac{2}{3}\times 0^{3}+5\times 0\right)$$

Simplify.

$$\frac{46}{3}$$