Calculus Examples

$\int_{0}^{10} | x - 5 | d x$

Step 1

Split up the integral depending on where $x - 5$ is positive and negative.

$\int_{0}^{5} - x + 5 d x + \int_{5}^{10} x - 5 d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{5} - x d x + \int_{0}^{5} 5 d x + \int_{5}^{10} x - 5 d x$

Step 3

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

$- \int_{0}^{5} x d x + \int_{0}^{5} 5 d x + \int_{5}^{10} x - 5 d x$

Step 4

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

$- (\frac{1}{2} x^{2}]_{0}^{5}) + \int_{0}^{5} 5 d x + \int_{5}^{10} x - 5 d x$

Step 5

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

$- (\frac{x^{2}}{2}]_{0}^{5}) + \int_{0}^{5} 5 d x + \int_{5}^{10} x - 5 d x$

Step 6

Apply the constant rule.

$- (\frac{x^{2}}{2}]_{0}^{5}) + 5 x]_{0}^{5} + \int_{5}^{10} x - 5 d x$

Step 7

Split the single integral into multiple integrals.

$- (\frac{x^{2}}{2}]_{0}^{5}) + 5 x]_{0}^{5} + \int_{5}^{10} x d x + \int_{5}^{10} - 5 d x$

Step 8

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

$- (\frac{x^{2}}{2}]_{0}^{5}) + 5 x]_{0}^{5} + \frac{1}{2} x^{2}]_{5}^{10} + \int_{5}^{10} - 5 d x$

Step 9

Apply the constant rule.

$- (\frac{x^{2}}{2}]_{0}^{5}) + 5 x]_{0}^{5} + \frac{1}{2} x^{2}]_{5}^{10} + - 5 x]_{5}^{10}$

Step 10

Simplify the answer.

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Step 10.1

Combine $\frac{1}{2} x^{2}]_{5}^{10}$ and $- 5 x]_{5}^{10}$ .

$- (\frac{x^{2}}{2}]_{0}^{5}) + 5 x]_{0}^{5} + \frac{1}{2} x^{2} - 5 x]_{5}^{10}$

Step 10.2

Substitute and simplify.

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Step 10.2.1

Evaluate $\frac{x^{2}}{2}$ at $5$ and at $0$ .

$- ((\frac{5^{2}}{2}) - \frac{0^{2}}{2}) + 5 x]_{0}^{5} + \frac{1}{2} x^{2} - 5 x]_{5}^{10}$

Step 10.2.2

Evaluate $5 x$ at $5$ and at $0$ .

$- (\frac{5^{2}}{2} - \frac{0^{2}}{2}) + 5 \cdot 5 - 5 \cdot 0 + \frac{1}{2} x^{2} - 5 x]_{5}^{10}$

Step 10.2.3

Evaluate $\frac{1}{2} x^{2} - 5 x$ at $10$ and at $5$ .

$- (\frac{5^{2}}{2} - \frac{0^{2}}{2}) + 5 \cdot 5 - 5 \cdot 0 + (\frac{1}{2} \cdot 10^{2} - 5 \cdot 10) - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4

Simplify.

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Step 10.2.4.1

Raise $5$ to the power of $2$ .

$- (\frac{25}{2} - \frac{0^{2}}{2}) + 5 \cdot 5 - 5 \cdot 0 + (\frac{1}{2} \cdot 10^{2} - 5 \cdot 10) - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.2

Raising $0$ to any positive power yields $0$ .

$- (\frac{25}{2} - \frac{0}{2}) + 5 \cdot 5 - 5 \cdot 0 + (\frac{1}{2} \cdot 10^{2} - 5 \cdot 10) - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.3

Cancel the common factor of $0$ and $2$ .

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Step 10.2.4.3.1

Factor $2$ out of $0$ .

$- (\frac{25}{2} - \frac{2 (0)}{2}) + 5 \cdot 5 - 5 \cdot 0 + (\frac{1}{2} \cdot 10^{2} - 5 \cdot 10) - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.3.2

Cancel the common factors.

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Step 10.2.4.3.2.1

Factor $2$ out of $2$ .

$- (\frac{25}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + 5 \cdot 5 - 5 \cdot 0 + (\frac{1}{2} \cdot 10^{2} - 5 \cdot 10) - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.3.2.2

Cancel the common factor.

$- (\frac{25}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + 5 \cdot 5 - 5 \cdot 0 + (\frac{1}{2} \cdot 10^{2} - 5 \cdot 10) - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.3.2.3

Rewrite the expression.

$- (\frac{25}{2} - \frac{0}{1}) + 5 \cdot 5 - 5 \cdot 0 + (\frac{1}{2} \cdot 10^{2} - 5 \cdot 10) - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.3.2.4

Divide $0$ by $1$ .

$- (\frac{25}{2} - 0) + 5 \cdot 5 - 5 \cdot 0 + (\frac{1}{2} \cdot 10^{2} - 5 \cdot 10) - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.4

Multiply $- 1$ by $0$ .

$- (\frac{25}{2} + 0) + 5 \cdot 5 - 5 \cdot 0 + (\frac{1}{2} \cdot 10^{2} - 5 \cdot 10) - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.5

Add $\frac{25}{2}$ and $0$ .

$- \frac{25}{2} + 5 \cdot 5 - 5 \cdot 0 + (\frac{1}{2} \cdot 10^{2} - 5 \cdot 10) - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.6

Multiply $5$ by $5$ .

$- \frac{25}{2} + 25 - 5 \cdot 0 + (\frac{1}{2} \cdot 10^{2} - 5 \cdot 10) - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.7

Multiply $- 5$ by $0$ .

$- \frac{25}{2} + 25 + 0 + (\frac{1}{2} \cdot 10^{2} - 5 \cdot 10) - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.8

Add $25$ and $0$ .

$- \frac{25}{2} + 25 + (\frac{1}{2} \cdot 10^{2} - 5 \cdot 10) - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.9

To write $25$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{25}{2} + 25 \cdot \frac{2}{2} + (\frac{1}{2} \cdot 10^{2} - 5 \cdot 10) - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.10

Combine $25$ and $\frac{2}{2}$ .

$- \frac{25}{2} + \frac{25 \cdot 2}{2} + (\frac{1}{2} \cdot 10^{2} - 5 \cdot 10) - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.11

Combine the numerators over the common denominator.

$\frac{- 25 + 25 \cdot 2}{2} + (\frac{1}{2} \cdot 10^{2} - 5 \cdot 10) - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.12

Simplify the numerator.

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Step 10.2.4.12.1

Multiply $25$ by $2$ .

$\frac{- 25 + 50}{2} + (\frac{1}{2} \cdot 10^{2} - 5 \cdot 10) - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.12.2

Add $- 25$ and $50$ .

$\frac{25}{2} + (\frac{1}{2} \cdot 10^{2} - 5 \cdot 10) - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.13

Raise $10$ to the power of $2$ .

$\frac{25}{2} + \frac{1}{2} \cdot 100 - 5 \cdot 10 - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.14

Combine $\frac{1}{2}$ and $100$ .

$\frac{25}{2} + \frac{100}{2} - 5 \cdot 10 - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.15

Cancel the common factor of $100$ and $2$ .

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Step 10.2.4.15.1

Factor $2$ out of $100$ .

$\frac{25}{2} + \frac{2 \cdot 50}{2} - 5 \cdot 10 - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.15.2

Cancel the common factors.

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Step 10.2.4.15.2.1

Factor $2$ out of $2$ .

$\frac{25}{2} + \frac{2 \cdot 50}{2 (1)} - 5 \cdot 10 - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.15.2.2

Cancel the common factor.

$\frac{25}{2} + \frac{2 \cdot 50}{2 \cdot 1} - 5 \cdot 10 - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.15.2.3

Rewrite the expression.

$\frac{25}{2} + \frac{50}{1} - 5 \cdot 10 - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.15.2.4

Divide $50$ by $1$ .

$\frac{25}{2} + 50 - 5 \cdot 10 - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.16

Multiply $- 5$ by $10$ .

$\frac{25}{2} + 50 - 50 - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.17

Subtract $50$ from $50$ .

$\frac{25}{2} + 0 - (\frac{1}{2} \cdot 5^{2} - 5 \cdot 5)$

Step 10.2.4.18

Raise $5$ to the power of $2$ .

$\frac{25}{2} + 0 - (\frac{1}{2} \cdot 25 - 5 \cdot 5)$

Step 10.2.4.19

Combine $\frac{1}{2}$ and $25$ .

$\frac{25}{2} + 0 - (\frac{25}{2} - 5 \cdot 5)$

Step 10.2.4.20

Multiply $- 5$ by $5$ .

$\frac{25}{2} + 0 - (\frac{25}{2} - 25)$

Step 10.2.4.21

To write $- 25$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{25}{2} + 0 - (\frac{25}{2} - 25 \cdot \frac{2}{2})$

Step 10.2.4.22

Combine $- 25$ and $\frac{2}{2}$ .

$\frac{25}{2} + 0 - (\frac{25}{2} + \frac{- 25 \cdot 2}{2})$

Step 10.2.4.23

Combine the numerators over the common denominator.

$\frac{25}{2} + 0 - \frac{25 - 25 \cdot 2}{2}$

Step 10.2.4.24

Simplify the numerator.

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Step 10.2.4.24.1

Multiply $- 25$ by $2$ .

$\frac{25}{2} + 0 - \frac{25 - 50}{2}$

Step 10.2.4.24.2

Subtract $50$ from $25$ .

$\frac{25}{2} + 0 - \frac{- 25}{2}$

Step 10.2.4.25

Move the negative in front of the fraction.

$\frac{25}{2} + 0 - - \frac{25}{2}$

Step 10.2.4.26

Multiply $- 1$ by $- 1$ .

$\frac{25}{2} + 0 + 1 (\frac{25}{2})$

Step 10.2.4.27

Multiply $\frac{25}{2}$ by $1$ .

$\frac{25}{2} + 0 + \frac{25}{2}$

Step 10.2.4.28

Add $0$ and $\frac{25}{2}$ .

$\frac{25}{2} + \frac{25}{2}$

Step 10.2.4.29

Combine the numerators over the common denominator.

$\frac{25 + 25}{2}$

Step 10.2.4.30

Add $25$ and $25$ .

$\frac{50}{2}$

Step 10.2.4.31

Cancel the common factor of $50$ and $2$ .

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Step 10.2.4.31.1

Factor $2$ out of $50$ .

$\frac{2 \cdot 25}{2}$

Step 10.2.4.31.2

Cancel the common factors.

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Step 10.2.4.31.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 25}{2 (1)}$

Step 10.2.4.31.2.2

Cancel the common factor.

$\frac{2 \cdot 25}{2 \cdot 1}$

Step 10.2.4.31.2.3

Rewrite the expression.

$\frac{25}{1}$

Step 10.2.4.31.2.4

Divide $25$ by $1$ .

$25$

Step 11