Calculus Examples

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

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

Step 1

Let

u = \ln (x)

$u = \ln (x)$ . Then

d u = \frac{1}{x} d x

$d u = \frac{1}{x} d x$ , so

x d u = d x

$x d u = d x$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

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Step 1.1

Let

u = \ln (x)

$u = \ln (x)$ . Find

\frac{d u}{d x}

$\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate

\ln (x)

$\ln (x)$ .

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

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

Step 1.1.2

The derivative of

\ln (x)

$\ln (x)$ with respect to

x

$x$ is

\frac{1}{x}

$\frac{1}{x}$ .

\frac{1}{x}

$\frac{1}{x}$

\frac{1}{x}

$\frac{1}{x}$

Step 1.2

Rewrite the problem using

u

$u$ and

d u

$d u$ .

\int u^{2} d u

$\int u^{2} d u$

\int u^{2} d u

$\int u^{2} d u$

Step 2

By the Power Rule, the integral of

u^{2}

$u^{2}$ with respect to

u

$u$ is

\frac{1}{3} u^{3}

$\frac{1}{3} u^{3}$ .

\frac{1}{3} u^{3} + C

$\frac{1}{3} u^{3} + C$

Step 3

Replace all occurrences of

u

$u$ with

\ln (x)

$\ln (x)$ .

\frac{1}{3} \ln^{3} (x) + C

$\frac{1}{3} \ln^{3} (x) + C$

[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$