Calculus Examples

$x \log (x)$

Step 1

Write $x \log (x)$ as a function.

$f (x) = x \log (x)$

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.

$F (x) = \int x \log (x) d x$

Step 4

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

$\log (x) (\frac{1}{2} x^{2}) - \int \frac{1}{2} x^{2} \frac{1}{x \ln (10)} d x$

Step 5

Simplify.

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Step 5.1

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

$\log (x) \frac{x^{2}}{2} - \int \frac{1}{2} x^{2} \frac{1}{x \ln (10)} d x$

Step 5.2

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

$\frac{\log (x) x^{2}}{2} - \int \frac{1}{2} x^{2} \frac{1}{x \ln (10)} d x$

Step 6

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

$\frac{\log (x) x^{2}}{2} - (\frac{1}{2} \int x^{2} \frac{1}{x \ln (10)} d x)$

Step 7

Simplify.

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Step 7.1

Combine $x^{2}$ and $\frac{1}{x \ln (10)}$ .

$\frac{\log (x) x^{2}}{2} - (\frac{1}{2} \int \frac{x^{2}}{x \ln (10)} d x)$

Step 7.2

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

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Step 7.2.1

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

$\frac{\log (x) x^{2}}{2} - (\frac{1}{2} \int \frac{x \cdot x}{x \ln (10)} d x)$

Step 7.2.2

Cancel the common factors.

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Step 7.2.2.1

Factor $x$ out of $x \ln (10)$ .

$\frac{\log (x) x^{2}}{2} - (\frac{1}{2} \int \frac{x \cdot x}{x (\ln (10))} d x)$

Step 7.2.2.2

Cancel the common factor.

$\frac{\log (x) x^{2}}{2} - (\frac{1}{2} \int \frac{x \cdot x}{x \ln (10)} d x)$

Step 7.2.2.3

Rewrite the expression.

$\frac{\log (x) x^{2}}{2} - (\frac{1}{2} \int \frac{x}{\ln (10)} d x)$

$\frac{\log (x) x^{2}}{2} - \frac{1}{2} \int \frac{x}{\ln (10)} d x$

Step 8

Since $\frac{1}{\ln (10)}$ is constant with respect to $x$ , move $\frac{1}{\ln (10)}$ out of the integral.

$\frac{\log (x) x^{2}}{2} - \frac{1}{2} (\frac{1}{\ln (10)} \int x d x)$

Step 9

Simplify.

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Step 9.1

Multiply $\frac{1}{\ln (10)}$ by $\frac{1}{2}$ .

$\frac{\log (x) x^{2}}{2} - \frac{1}{\ln (10) \cdot 2} \int x d x$

Step 9.2

Move $2$ to the left of $\ln (10)$ .

$\frac{\log (x) x^{2}}{2} - \frac{1}{2 \ln (10)} \int x d x$

Step 10

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

$\frac{\log (x) x^{2}}{2} - \frac{1}{2 \ln (10)} (\frac{1}{2} x^{2} + C)$

Step 11

Simplify the answer.

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Step 11.1

Rewrite $\frac{\log (x) x^{2}}{2} - \frac{1}{2 \ln (10)} (\frac{1}{2} x^{2} + C)$ as $\frac{1}{2} \log (x) x^{2} - \frac{1}{2 \ln (10)} \cdot \frac{1}{2} x^{2} + C$ .

$\frac{1}{2} \log (x) x^{2} - \frac{1}{2 \ln (10)} \cdot \frac{1}{2} x^{2} + C$

Step 11.2

Simplify.

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Step 11.2.1

Combine $\frac{1}{2}$ and $\log (x)$ .

$\frac{\log (x)}{2} x^{2} - \frac{1}{2 \ln (10)} \cdot \frac{1}{2} x^{2} + C$

Step 11.2.2

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

$\frac{\log (x) x^{2}}{2} - \frac{1}{2 \ln (10)} \cdot \frac{1}{2} x^{2} + C$

Step 11.2.3

Multiply $\frac{1}{2}$ by $\frac{1}{2 \ln (10)}$ .

$\frac{\log (x) x^{2}}{2} - \frac{1}{2 (2 \ln (10))} x^{2} + C$

Step 11.2.4

Multiply $2$ by $2$ .

$\frac{\log (x) x^{2}}{2} - \frac{1}{4 \ln (10)} x^{2} + C$

Step 11.3

Combine $x^{2}$ and $\frac{1}{4 \ln (10)}$ .

$\frac{\log (x) x^{2}}{2} - \frac{x^{2}}{4 \ln (10)} + C$

Step 11.4

Reorder terms.

$\frac{1}{2} \log (x) x^{2} - \frac{1}{4 \ln (10)} x^{2} + C$

Step 12

The answer is the antiderivative of the function $f (x) = x \log (x)$ .

$F (x) =$ $\frac{1}{2} \log (x) x^{2} - \frac{1}{4 \ln (10)} x^{2} + C$