Calculus Examples

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

Step 1

Split the fraction into multiple fractions.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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 4.1

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

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Step 4.1.1

Differentiate $\ln (x)$ .

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

Step 4.1.2

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

$\frac{1}{x}$

Step 4.2

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

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

Step 5

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 6

Simplify.

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

Step 7

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

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