Solve ∫ (from 0.5 to 1) of 2/3+x^2 wrt x

$$\int_{ 0.5 }^{ 1 } \frac{ 2 }{ 3 } + { x }^{ 2 } d x$$

Evaluate

$$0.625$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Find the integral of \frac{2}{3} using the table of common integrals rule \int a\mathrm{d}x=ax.

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

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}.

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

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.

$$\frac{2}{3}\times 1+\frac{1^{3}}{3}-\left(\frac{2}{3}\times 0.5+\frac{0.5^{3}}{3}\right)$$

Simplify.

$$\frac{5}{8}$$