Calculus Examples

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

Step 1

Write $\frac{\ln (x^{2})}{x}$ as a function.

$f (x) = \frac{\ln (x^{2})}{x}$

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.

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

Step 4

Rewrite $\frac{\ln (x^{2})}{x}$ as $\frac{2 \ln (x)}{x}$ .

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

Step 5

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 6

Let $u = \ln (x)$ . Then $d u = \frac{1}{x} d x$ , so $x d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.1

Let $u = \ln (x)$ . Find $\frac{d u}{d x}$ .

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Step 6.1.1

Differentiate $\ln (x)$ .

$\frac{d}{d x} [\ln (x)]$

Step 6.1.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$\frac{1}{x}$

Step 6.2

Rewrite the problem using $u$ and $d u$ .

$2 \int u d u$

Step 7

By the Power Rule, the integral of $u$ with respect to $u$ is $\frac{1}{2} u^{2}$ .

$2 (\frac{1}{2} u^{2} + C)$

Step 8

Simplify.

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Step 8.1

Rewrite $2 (\frac{1}{2} u^{2} + C)$ as $2 (\frac{1}{2}) u^{2} + C$ .

$2 (\frac{1}{2}) u^{2} + C$

Step 8.2

Simplify.

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Step 8.2.1

Combine $2$ and $\frac{1}{2}$ .

$\frac{2}{2} u^{2} + C$

Step 8.2.2

Cancel the common factor of $2$ .

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Step 8.2.2.1

Cancel the common factor.

$\frac{2}{2} u^{2} + C$

Step 8.2.2.2

Rewrite the expression.

$1 u^{2} + C$

Step 8.2.3

Multiply $u^{2}$ by $1$ .

$u^{2} + C$

Step 9

Replace all occurrences of $u$ with $\ln (x)$ .

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

Step 10

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

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