Calculus Examples

$\log_{2} (x)$

Step 1

Write $\log_{2} (x)$ as a function.

$f (x) = \log_{2} (x)$

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.

$F (x) = \int \log_{2} (x) d x$

Step 4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \log_{2} (x)$ and $d v = 1$ .

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

Step 5

Simplify.

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Step 5.1

Combine $x$ and $\frac{1}{x \ln (2)}$ .

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

Step 5.2

Cancel the common factor of $x$ .

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Rewrite the expression.

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

Step 6

Apply the constant rule.

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

Step 7

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

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

Step 8

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

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