Solve ∫ (from 0 to 2) of e^x+2x wrt x

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

Evaluate

$$e^{2}+3\approx 10.389056099$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

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

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply 2 times \frac{x^{2}}{2}.

$$e^{x}+x^{2}$$

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.

$$e^{2}+2^{2}-\left(e^{0}+0^{2}\right)$$

Simplify.

$$e^{2}+3$$