Calculus Examples

$\int_{10}^{20} \frac{30}{x} + 25 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{10}^{20} \frac{30}{x} d x + \int_{10}^{20} 25 d x$

Step 2

Since $30$ is constant with respect to $x$ , move $30$ out of the integral.

$30 \int_{10}^{20} \frac{1}{x} d x + \int_{10}^{20} 25 d x$

Step 3

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

$30 (\ln (| x |)]_{10}^{20}) + \int_{10}^{20} 25 d x$

Step 4

Apply the constant rule.

$30 (\ln (| x |)]_{10}^{20}) + 25 x]_{10}^{20}$

Step 5

Simplify the answer.

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Step 5.1

Substitute and simplify.

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Step 5.1.1

Evaluate $\ln (| x |)$ at $20$ and at $10$ .

$30 ((\ln (| 20 |)) - \ln (| 10 |)) + 25 x]_{10}^{20}$

Step 5.1.2

Evaluate $25 x$ at $20$ and at $10$ .

$30 (\ln (| 20 |) - \ln (| 10 |)) + 25 \cdot 20 - 25 \cdot 10$

Step 5.1.3

Simplify.

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Step 5.1.3.1

Multiply $25$ by $20$ .

$30 (\ln (| 20 |) - \ln (| 10 |)) + 500 - 25 \cdot 10$

Step 5.1.3.2

Multiply $- 25$ by $10$ .

$30 (\ln (| 20 |) - \ln (| 10 |)) + 500 - 250$

Step 5.1.3.3

Subtract $250$ from $500$ .

$30 (\ln (| 20 |) - \ln (| 10 |)) + 250$

Step 5.2

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

$30 \ln (\frac{| 20 |}{| 10 |}) + 250$

Step 5.3

Simplify.

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Step 5.3.1

The absolute value is the distance between a number and zero. The distance between $0$ and $20$ is $20$ .

$30 \ln (\frac{20}{| 10 |}) + 250$

Step 5.3.2

The absolute value is the distance between a number and zero. The distance between $0$ and $10$ is $10$ .

$30 \ln (\frac{20}{10}) + 250$

Step 5.3.3

Divide $20$ by $10$ .

$30 \ln (2) + 250$

Step 6

The result can be shown in multiple forms.

Exact Form:

$30 \ln (2) + 250$

Decimal Form:

$270.79441541 \dots$

Step 7