Calculus Examples

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

Step 1

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

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

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.

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

Step 4

Since $\frac{1}{\ln (2)}$ is constant with respect to $x$ , move $\frac{1}{\ln (2)}$ out of the integral.

$\frac{1}{\ln (2)} \int 2^{x} d x$

Step 5

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

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

Step 6

Simplify the answer.

Tap for more steps...

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

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

Simplify.

Tap for more steps...

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

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

Raise $\ln (2)$ to the power of $1$ .

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

Raise $\ln (2)$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Step 7

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

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