Calculus Examples

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

Step 1

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 2

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 3

Simplify the answer.

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Step 3.1

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

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

Step 3.2

Simplify.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.2.5

Add $1$ and $1$ .

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