Calculus Examples

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

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln^{2} (x)$ and $d v = x$ .

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Rewrite $\frac{1}{2} x^{2} \frac{\ln (x^{2})}{x}$ as $\frac{1}{2} x^{2} \frac{2 \ln (x)}{x}$ .

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

Factor $x$ out of $x^{2} (2 \ln (x))$ .

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

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Raise $x$ to the power of $1$ .

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

Step 5.2.2.2

Factor $x$ out of $x^{1}$ .

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

Step 5.2.2.3

Cancel the common factor.

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

Step 5.2.2.4

Rewrite the expression.

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

Step 5.2.2.5

Divide $x (2 \ln (x))$ by $1$ .

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

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Multiply $2$ by $- 1$ .

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

Step 7.2

Combine $- 2$ and $\frac{1}{2}$ .

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

Step 7.3

Cancel the common factor of $- 2$ and $2$ .

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Step 7.3.1

Factor $2$ out of $- 2$ .

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

Step 7.3.2

Cancel the common factors.

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Step 7.3.2.1

Factor $2$ out of $2$ .

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

Step 7.3.2.2

Cancel the common factor.

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

Step 7.3.2.3

Rewrite the expression.

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

Step 7.3.2.4

Divide $- 1$ by $1$ .

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

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

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

Step 9.5

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

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Step 9.5.1

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

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

Step 9.5.2

Cancel the common factors.

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Step 9.5.2.1

Factor $x$ out of $2 x$ .

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

Step 9.5.2.2

Cancel the common factor.

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

Step 9.5.2.3

Rewrite the expression.

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

Step 10

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

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

Step 11

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

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

Step 12

Simplify the answer.

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Step 12.1

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

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

Step 12.2

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

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

Step 12.3

Simplify.

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Step 12.3.1

To write $- (\frac{\ln (x) x^{2}}{2} - \frac{x^{2}}{4})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 12.3.2

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

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

Step 12.3.3

Combine the numerators over the common denominator.

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

Step 12.3.4

Multiply $2$ by $- 1$ .

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

Step 12.4

Simplify.

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Step 12.4.1

Apply the distributive property.

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

Step 12.4.2

Cancel the common factor of $2$ .

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Step 12.4.2.1

Factor $2$ out of $- 2$ .

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

Step 12.4.2.2

Cancel the common factor.

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

Step 12.4.2.3

Rewrite the expression.

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

Step 12.4.3

Cancel the common factor of $2$ .

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Step 12.4.3.1

Move the leading negative in $- \frac{x^{2}}{4}$ into the numerator.

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

Step 12.4.3.2

Factor $2$ out of $- 2$ .

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

Step 12.4.3.3

Factor $2$ out of $4$ .

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

Step 12.4.3.4

Cancel the common factor.

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

Step 12.4.3.5

Rewrite the expression.

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

Step 12.4.4

Simplify each term.

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Step 12.4.4.1

Move the negative in front of the fraction.

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

Step 12.4.4.2

Multiply $- - \frac{x^{2}}{2}$ .

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Step 12.4.4.2.1

Multiply $- 1$ by $- 1$ .

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

Step 12.4.4.2.2

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

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

Step 12.4.5

To write $- \ln (x) x^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 12.4.6

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

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

Step 12.4.7

Combine the numerators over the common denominator.

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

Step 12.4.8

Simplify the numerator.

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Step 12.4.8.1

Factor $x^{2}$ out of $- \ln (x) x^{2} \cdot 2 + x^{2}$ .

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Step 12.4.8.1.1

Factor $x^{2}$ out of $- \ln (x) x^{2} \cdot 2$ .

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

Step 12.4.8.1.2

Multiply by $1$ .

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

Step 12.4.8.1.3

Factor $x^{2}$ out of $x^{2} (- \ln (x) \cdot 2) + x^{2} \cdot 1$ .

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

Step 12.4.8.2

Multiply $2$ by $- 1$ .

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

Step 12.4.9

Factor $- 1$ out of $- 2 \ln (x)$ .

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

Step 12.4.10

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 12.4.11

Factor $- 1$ out of $- (2 \ln (x)) - 1 (- 1)$ .

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

Step 12.4.12

Rewrite $- (2 \ln (x) - 1)$ as $- 1 (2 \ln (x) - 1)$ .

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

Step 12.4.13

Move the negative in front of the fraction.

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

Step 12.5

Reorder terms.

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