Calculus Examples

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

Step 1

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

$\int 2 x (2 \ln (x)) d x$

Step 2

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

$2 \cdot 2 \int x (\ln (x)) d x$

Step 3

Multiply $2$ by $2$ .

$4 \int x (\ln (x)) d x$

Step 4

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

$4 (\ln (x) (\frac{1}{2} x^{2}) - \int \frac{1}{2} x^{2} \frac{1}{x} d x)$

Step 5

Simplify.

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Step 5.1

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

$4 (\ln (x) \frac{x^{2}}{2} - \int \frac{1}{2} x^{2} \frac{1}{x} d x)$

Step 5.2

Combine $\ln (x)$ and $\frac{x^{2}}{2}$ .

$4 (\frac{\ln (x) x^{2}}{2} - \int \frac{1}{2} x^{2} \frac{1}{x} d x)$

Step 5.3

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

$4 (\frac{\ln (x) x^{2}}{2} - \int \frac{x^{2}}{2} \cdot \frac{1}{x} d x)$

Step 5.4

Multiply $\frac{x^{2}}{2}$ by $\frac{1}{x}$ .

$4 (\frac{\ln (x) x^{2}}{2} - \int \frac{x^{2}}{2 x} d x)$

Step 5.5

Cancel the common factor of $x^{2}$ and $x$ .

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Step 5.5.1

Factor $x$ out of $x^{2}$ .

$4 (\frac{\ln (x) x^{2}}{2} - \int \frac{x \cdot x}{2 x} d x)$

Step 5.5.2

Cancel the common factors.

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Step 5.5.2.1

Factor $x$ out of $2 x$ .

$4 (\frac{\ln (x) x^{2}}{2} - \int \frac{x \cdot x}{x \cdot 2} d x)$

Step 5.5.2.2

Cancel the common factor.

$4 (\frac{\ln (x) x^{2}}{2} - \int \frac{x \cdot x}{x \cdot 2} d x)$

Step 5.5.2.3

Rewrite the expression.

$4 (\frac{\ln (x) x^{2}}{2} - \int \frac{x}{2} d x)$

Step 6

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$4 (\frac{\ln (x) x^{2}}{2} - (\frac{1}{2} \int x d x))$

Step 7

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

$4 (\frac{\ln (x) x^{2}}{2} - \frac{1}{2} (\frac{1}{2} x^{2} + C))$

Step 8

Simplify the answer.

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Step 8.1

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

$4 (\frac{\ln (x) x^{2}}{2} - \frac{1}{2} (\frac{x^{2}}{2} + C))$

Step 8.2

Rewrite $4 (\frac{\ln (x) x^{2}}{2} - \frac{1}{2} (\frac{x^{2}}{2} + C))$ as $4 (\frac{\ln (x) x^{2}}{2} - \frac{x^{2}}{4}) + C$ .

$4 (\frac{\ln (x) x^{2}}{2} - \frac{x^{2}}{4}) + C$

Step 8.3

Reorder terms.

$4 (\frac{1}{2} \ln (x) x^{2} - \frac{1}{4} x^{2}) + C$