Calculus Examples

$\ln (x d x)$

Step 1

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

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

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.

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

Step 4

Simplify.

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Step 4.1

Raise $x$ to the power of $1$ .

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

Step 4.2

Raise $x$ to the power of $1$ .

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

Step 4.3

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

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

Step 4.4

Add $1$ and $1$ .

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Combine $x$ and $\frac{2}{x}$ .

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

Step 6.2

Move $2$ to the left of $x$ .

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

Step 6.3

Cancel the common factor of $x$ .

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Step 6.3.1

Cancel the common factor.

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

Step 6.3.2

Divide $2$ by $1$ .

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

Step 7

Apply the constant rule.

$\ln (d x^{2}) x - (2 x + C)$

Step 8

Rewrite $\ln (d x^{2}) x - (2 x + C)$ as $\ln (d x^{2}) x - 2 x + C$ .

$\ln (d x^{2}) x - 2 x + C$

Step 9

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

$F (x) =$ $\ln (d x^{2}) x - 2 x + C$