Calculus Examples

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

Step 1

Remove parentheses.

$\int_{1}^{2} \frac{x}{2} - \frac{2}{x} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{1}^{2} \frac{x}{2} d x + \int_{1}^{2} - \frac{2}{x} d x$

Step 3

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

$\frac{1}{2} \int_{1}^{2} x d x + \int_{1}^{2} - \frac{2}{x} d x$

Step 4

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

$\frac{1}{2} \frac{1}{2} x^{2}]_{1}^{2} + \int_{1}^{2} - \frac{2}{x} d x$

Step 5

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 6

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

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

Step 7

Multiply $2$ by $- 1$ .

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

Step 8

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

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

Step 9

Simplify the answer.

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Step 9.1

Substitute and simplify.

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Step 9.1.1

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

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

Step 9.1.2

Evaluate $\ln (| x |)$ at $2$ and at $1$ .

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

Step 9.1.3

Simplify.

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Step 9.1.3.1

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

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

Step 9.1.3.2

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

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

Step 9.1.3.3

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

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Step 9.1.3.3.1

Factor $2$ out of $4$ .

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

Step 9.1.3.3.2

Cancel the common factors.

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Step 9.1.3.3.2.1

Factor $2$ out of $2$ .

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

Step 9.1.3.3.2.2

Cancel the common factor.

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

Step 9.1.3.3.2.3

Rewrite the expression.

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

Step 9.1.3.3.2.4

Divide $2$ by $1$ .

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

Step 9.1.3.4

One to any power is one.

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

Step 9.1.3.5

Multiply $- 1$ by $1$ .

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

Step 9.1.3.6

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

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

Step 9.1.3.7

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

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

Step 9.1.3.8

Combine the numerators over the common denominator.

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

Step 9.1.3.9

Simplify the numerator.

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Step 9.1.3.9.1

Multiply $2$ by $2$ .

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

Step 9.1.3.9.2

Subtract $1$ from $4$ .

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

Step 9.1.3.10

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

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

Step 9.1.3.11

Multiply $2$ by $2$ .

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

Step 9.1.3.12

To write $- 2 (\ln (| 2 |) - \ln (| 1 |))$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 9.1.3.13

Combine $- 2 (\ln (| 2 |) - \ln (| 1 |))$ and $\frac{4}{4}$ .

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

Step 9.1.3.14

Combine the numerators over the common denominator.

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

Step 9.1.3.15

Multiply $4$ by $- 2$ .

$\frac{3 - 8 (\ln (| 2 |) - \ln (| 1 |))}{4}$

Step 9.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{3 - 8 \ln (\frac{| 2 |}{| 1 |})}{4}$

Step 9.3

Simplify.

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Step 9.3.1

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{3 - 8 \ln (\frac{2}{| 1 |})}{4}$

Step 9.3.2

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{3 - 8 \ln (\frac{2}{1})}{4}$

Step 9.3.3

Divide $2$ by $1$ .

$\frac{3 - 8 \ln (2)}{4}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{3 - 8 \ln (2)}{4}$

Decimal Form:

$- 0.63629436 \dots$

Step 11