Calculus Examples

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

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

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

Simplify terms.

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Write $x^{2}$ as a fraction with denominator $1$ .

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

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

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

Write $\ln (x)$ as a fraction with denominator $1$ .

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

Multiply $\frac{\ln (x)}{1}$ and $\frac{x^{2}}{2}$ .

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

Write $x^{2}$ as a fraction with denominator $1$ .

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

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

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

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

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

Reduce the expression by cancelling the common factors.

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Reorder $2$ and $x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Remove parentheses around $\frac{1}{2} \int x d x$ .

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

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

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

Simplify the answer.

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Write $x^{2}$ as a fraction with denominator $1$ .

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

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

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

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

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

Reorder terms.

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

Remove parentheses around $\frac{1}{4} x^{2}$ .

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

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