Algebra Examples

$\ln (2 x)$

Step 1

Write $\ln (2 x)$ as a function.

$f (x) = \ln (2 x)$

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.

$F (x) = \int \ln (2 x) d x$

Step 4

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

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

Step 5

Simplify.

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Combine $x$ and $\frac{1}{x}$ .

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

Cancel the common factor of $x$ .

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Cancel the common factor.

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

Rewrite the expression.

$\ln (2 x) x - \int d x$

Step 6

Apply the constant rule.

$\ln (2 x) x - (x + C)$

Step 7

Simplify.

$\ln (2 x) x - x + C$

Step 8

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

$F (x) =$ $\ln (2 x) x - x + C$