Calculus Examples

$\int_{1}^{2} x \ln (x) d x$

Step 1

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

$\ln (x) (\frac{1}{2} x^{2})]_{1}^{2} - \int_{1}^{2} \frac{1}{2} x^{2} \frac{1}{x} d x$

Step 2

Simplify.

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Step 2.1

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

$\ln (x) \frac{x^{2}}{2}]_{1}^{2} - \int_{1}^{2} \frac{1}{2} x^{2} \frac{1}{x} d x$

Step 2.2

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

$\frac{\ln (x) x^{2}}{2}]_{1}^{2} - \int_{1}^{2} \frac{1}{2} x^{2} \frac{1}{x} d x$

Step 3

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

$\frac{\ln (x) x^{2}}{2}]_{1}^{2} - (\frac{1}{2} \int_{1}^{2} x^{2} \frac{1}{x} d x)$

Step 4

Simplify.

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Step 4.1

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

$\frac{\ln (x) x^{2}}{2}]_{1}^{2} - (\frac{1}{2} \int_{1}^{2} \frac{x^{2}}{x} d x)$

Step 4.2

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

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Step 4.2.1

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

$\frac{\ln (x) x^{2}}{2}]_{1}^{2} - (\frac{1}{2} \int_{1}^{2} \frac{x \cdot x}{x} d x)$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

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

$\frac{\ln (x) x^{2}}{2}]_{1}^{2} - (\frac{1}{2} \int_{1}^{2} \frac{x \cdot x}{x^{1}} d x)$

Step 4.2.2.2

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

$\frac{\ln (x) x^{2}}{2}]_{1}^{2} - (\frac{1}{2} \int_{1}^{2} \frac{x \cdot x}{x \cdot 1} d x)$

Step 4.2.2.3

Cancel the common factor.

$\frac{\ln (x) x^{2}}{2}]_{1}^{2} - (\frac{1}{2} \int_{1}^{2} \frac{x \cdot x}{x \cdot 1} d x)$

Step 4.2.2.4

Rewrite the expression.

$\frac{\ln (x) x^{2}}{2}]_{1}^{2} - (\frac{1}{2} \int_{1}^{2} \frac{x}{1} d x)$

Step 4.2.2.5

Divide $x$ by $1$ .

$\frac{\ln (x) x^{2}}{2}]_{1}^{2} - (\frac{1}{2} \int_{1}^{2} x d x)$

$\frac{\ln (x) x^{2}}{2}]_{1}^{2} - \frac{1}{2} \int_{1}^{2} x d x$

Step 5

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

$\frac{\ln (x) x^{2}}{2}]_{1}^{2} - \frac{1}{2} \frac{1}{2} x^{2}]_{1}^{2}$

Step 6

Substitute and simplify.

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Step 6.1

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

$(\frac{\ln (2) \cdot 2^{2}}{2}) - \frac{\ln (1) \cdot 1^{2}}{2} - \frac{1}{2} \frac{1}{2} x^{2}]_{1}^{2}$

Step 6.2

Evaluate $\frac{1}{2} x^{2}$ at $2$ and at $1$ .

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

Step 6.3

Simplify.

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Step 6.3.1

Raise $2$ to the power of $2$ .

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

Step 6.3.2

Move $4$ to the left of $\ln (2)$ .

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

Step 6.3.3

Cancel the common factor of $4$ and $2$ .

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Step 6.3.3.1

Factor $2$ out of $4 \ln (2)$ .

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

Step 6.3.3.2

Cancel the common factors.

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Step 6.3.3.2.1

Factor $2$ out of $2$ .

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

Step 6.3.3.2.2

Cancel the common factor.

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

Step 6.3.3.2.3

Rewrite the expression.

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

Step 6.3.3.2.4

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

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

Step 6.3.4

One to any power is one.

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

Step 6.3.5

Multiply $\ln (1)$ by $1$ .

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

Step 6.3.6

Raise $2$ to the power of $2$ .

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

Step 6.3.7

Combine $\frac{1}{2}$ and $4$ .

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

Step 6.3.8

Cancel the common factor of $4$ and $2$ .

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Step 6.3.8.1

Factor $2$ out of $4$ .

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

Step 6.3.8.2

Cancel the common factors.

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Step 6.3.8.2.1

Factor $2$ out of $2$ .

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

Step 6.3.8.2.2

Cancel the common factor.

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

Step 6.3.8.2.3

Rewrite the expression.

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

Step 6.3.8.2.4

Divide $2$ by $1$ .

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

Step 6.3.9

One to any power is one.

$2 \ln (2) - \frac{\ln (1)}{2} - \frac{1}{2} (2 - \frac{1}{2} \cdot 1)$

Step 6.3.10

Multiply $- 1$ by $1$ .

$2 \ln (2) - \frac{\ln (1)}{2} - \frac{1}{2} (2 - \frac{1}{2})$

Step 6.3.11

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 \ln (2) - \frac{\ln (1)}{2} - \frac{1}{2} (2 \cdot \frac{2}{2} - \frac{1}{2})$

Step 6.3.12

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

$2 \ln (2) - \frac{\ln (1)}{2} - \frac{1}{2} (\frac{2 \cdot 2}{2} - \frac{1}{2})$

Step 6.3.13

Combine the numerators over the common denominator.

$2 \ln (2) - \frac{\ln (1)}{2} - \frac{1}{2} \cdot \frac{2 \cdot 2 - 1}{2}$

Step 6.3.14

Simplify the numerator.

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Step 6.3.14.1

Multiply $2$ by $2$ .

$2 \ln (2) - \frac{\ln (1)}{2} - \frac{1}{2} \cdot \frac{4 - 1}{2}$

Step 6.3.14.2

Subtract $1$ from $4$ .

$2 \ln (2) - \frac{\ln (1)}{2} - \frac{1}{2} \cdot \frac{3}{2}$

Step 6.3.15

Multiply $\frac{3}{2}$ by $\frac{1}{2}$ .

$2 \ln (2) - \frac{\ln (1)}{2} - \frac{3}{2 \cdot 2}$

Step 6.3.16

Multiply $2$ by $2$ .

$2 \ln (2) - \frac{\ln (1)}{2} - \frac{3}{4}$

Step 7

Simplify.

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Step 7.1

Simplify each term.

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Step 7.1.1

The natural logarithm of $1$ is $0$ .

$2 \ln (2) - \frac{0}{2} - \frac{3}{4}$

Step 7.1.2

Divide $0$ by $2$ .

$2 \ln (2) - 0 - \frac{3}{4}$

Step 7.1.3

Multiply $- 1$ by $0$ .

$2 \ln (2) + 0 - \frac{3}{4}$

Step 7.2

Add $2 \ln (2)$ and $0$ .

$2 \ln (2) - \frac{3}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$2 \ln (2) - \frac{3}{4}$

Decimal Form:

$0.63629436 \dots$