Solve ∫ (from 0 to ln 2) of (1+e^x) wrt x

$$\int _ { 0 } ^ { \ln 2 } ( 1 + e ^ { x } ) d x$$

Evaluate

$$\ln(2)+1\approx 1.693147181$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 1+e^{x}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 1\mathrm{d}x+\int e^{x}\mathrm{d}x$$

Find the integral of 1 using the table of common integrals rule \int a\mathrm{d}x=ax.

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

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

$$x+e^{x}$$

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.

$$\ln(2)\ln(e)^{-1}+e^{\ln(2)\ln(e)^{-1}}-\left(0+e^{0}\right)$$

Simplify.

$$\ln(2)+1$$