Solve ∫ (from 0 to 3) of x^2+e^x

$$\int _ { 0 } ^ { 3 } x ^ { 2 } + e ^ { x }$$

Evaluate

$$e^{3}+8\approx 28.085536923$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

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}.

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

Use \int e^{x}\mathrm{d}x=e^{x} from the table of common integrals to obtain the result.

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

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.

$$\frac{3^{3}}{3}+e^{3}-\left(\frac{0^{3}}{3}+e^{0}\right)$$

Simplify.

$$8+e^{3}$$