Calculus Examples

$\frac{1 + \ln (x)}{x}$

Step 1

Write $\frac{1 + \ln (x)}{x}$ as a function.

$f (x) = \frac{1 + \ln (x)}{x}$

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.

$F (x) = \int \frac{1 + \ln (x)}{x} d x$

Step 4

Split the fraction into multiple fractions.

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

$\ln (| x |) + C + \int \frac{\ln (x)}{x} d x$

Step 7

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 7.1

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

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Step 7.1.1

Differentiate $\ln (x)$ .

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

Step 7.1.2

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

$\frac{1}{x}$

Step 7.2

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

$\ln (| x |) + C + \int u d u$

Step 8

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

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

Step 9

Simplify.

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

Step 10

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

$\ln (| x |) + \frac{1}{2} \ln^{2} (x) + C$

Step 11

The answer is the antiderivative of the function $f (x) = \frac{1 + \ln (x)}{x}$ .

$F (x) =$ $\ln (| x |) + \frac{1}{2} \ln^{2} (x) + C$