Solve ∫ (from 1 to 3) of (5^x+e^x) wrt x

$$\int _ { 1 } ^ { 3 } ( 5 ^ { x } + e ^ { x } ) d x$$

Evaluate

$$\frac{120}{\ln(5)}+e^{3}-e\approx 91.927447242$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Use \int x^{y}\mathrm{d}y=\frac{x^{y}}{\ln(x)} from the table of common integrals to obtain the result.

$$\frac{5^{x}}{\ln(5)}+\int e^{x}\mathrm{d}x$$

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

$$\frac{5^{x}}{\ln(5)}+e^{x}$$

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.

$$5^{3}\ln(5)^{-1}+e^{3}-\left(5^{1}\ln(5)^{-1}+e^{1}\right)$$

Simplify.

$$e^{3}+\frac{120}{\ln(5)}-e$$