Calculus Examples

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

Step 1

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 2

Simplify.

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Step 2.1

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

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

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 3

Apply the constant rule.

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

Step 4

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

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