Calculus Examples

${(\frac{2}{e})}^{x}$

Step 1

Write ${(\frac{2}{e})}^{x}$ as a function.

$f (x) = {(\frac{2}{e})}^{x}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int {(\frac{2}{e})}^{x} d x$

Step 4

The integral of ${(\frac{2}{e})}^{x}$ with respect to $x$ is $\frac{{(\frac{2}{e})}^{x}}{\ln (\frac{2}{e})}$ .

$\frac{{(\frac{2}{e})}^{x}}{\ln (\frac{2}{e})} + C$

Step 5

Simplify the answer.

Tap for more steps...

Step 5.1

Apply the product rule to $\frac{2}{e}$ .

$\frac{\frac{2^{x}}{e^{x}}}{\ln (\frac{2}{e})} + C$

Step 5.2

Simplify.

Tap for more steps...

Step 5.2.1

Rewrite $\frac{\frac{2^{x}}{e^{x}}}{\ln (\frac{2}{e})}$ as a product.

$\frac{2^{x}}{e^{x}} \cdot \frac{1}{\ln (\frac{2}{e})} + C$

Step 5.2.2

Multiply $\frac{2^{x}}{e^{x}}$ by $\frac{1}{\ln (\frac{2}{e})}$ .

$\frac{2^{x}}{e^{x} \ln (\frac{2}{e})} + C$

Step 5.3

Rewrite $\frac{2^{x}}{e^{x} \ln (\frac{2}{e})} + C$ as $\frac{2^{x}}{e^{x} \ln (2 e^{- 1})} + C$ .

$\frac{2^{x}}{e^{x} \ln (2 e^{- 1})} + C$

Step 6

The answer is the antiderivative of the function $f (x) = {(\frac{2}{e})}^{x}$ .

$F (x) =$ $\frac{2^{x}}{e^{x} \ln (2 e^{- 1})} + C$