Calculus Examples

$\int_{2}^{5} (2 - x) (x - 5) d x$

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

$\int_{2}^{5} 2 (x - 5) - x (x - 5) d x$

Step 1.2

Apply the distributive property.

$\int_{2}^{5} 2 x + 2 \cdot - 5 - x (x - 5) d x$

Step 1.3

Apply the distributive property.

$\int_{2}^{5} 2 x + 2 \cdot - 5 - x \cdot x - x \cdot - 5 d x$

Step 1.4

Move $x$ .

$\int_{2}^{5} 2 x + 2 \cdot - 5 - x \cdot x - 1 \cdot - 5 x d x$

Step 1.5

Multiply $2$ by $- 5$ .

$\int_{2}^{5} 2 x - 10 - x \cdot x - 1 \cdot - 5 x d x$

Step 1.6

Factor out negative.

$\int_{2}^{5} 2 x - 10 - (x \cdot x) - 1 \cdot - 5 x d x$

Step 1.7

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

$\int_{2}^{5} 2 x - 10 - (x^{1} x) - 1 \cdot - 5 x d x$

Step 1.8

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

$\int_{2}^{5} 2 x - 10 - (x^{1} x^{1}) - 1 \cdot - 5 x d x$

Step 1.9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int_{2}^{5} 2 x - 10 - x^{1 + 1} - 1 \cdot - 5 x d x$

Step 1.10

Add $1$ and $1$ .

$\int_{2}^{5} 2 x - 10 - x^{2} - 1 \cdot - 5 x d x$

Step 1.11

Multiply $- 1$ by $- 5$ .

$\int_{2}^{5} 2 x - 10 - x^{2} + 5 x d x$

Step 1.12

Move $- 10$ .

$\int_{2}^{5} 2 x - x^{2} + 5 x - 10 d x$

Step 1.13

Move $2 x$ .

$\int_{2}^{5} - x^{2} + 5 x + 2 x - 10 d x$

Step 1.14

Add $5 x$ and $2 x$ .

$\int_{2}^{5} - x^{2} + 7 x - 10 d x$

Step 2

Split the single integral into multiple integrals.

$\int_{2}^{5} - x^{2} d x + \int_{2}^{5} 7 x d x + \int_{2}^{5} - 10 d x$

Step 3

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

$- \int_{2}^{5} x^{2} d x + \int_{2}^{5} 7 x d x + \int_{2}^{5} - 10 d x$

Step 4

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$- (\frac{1}{3} x^{3}]_{2}^{5}) + \int_{2}^{5} 7 x d x + \int_{2}^{5} - 10 d x$

Step 5

Combine $\frac{1}{3}$ and $x^{3}$ .

$- (\frac{x^{3}}{3}]_{2}^{5}) + \int_{2}^{5} 7 x d x + \int_{2}^{5} - 10 d x$

Step 6

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$- (\frac{x^{3}}{3}]_{2}^{5}) + 7 \int_{2}^{5} x d x + \int_{2}^{5} - 10 d x$

Step 7

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

$- (\frac{x^{3}}{3}]_{2}^{5}) + 7 (\frac{1}{2} x^{2}]_{2}^{5}) + \int_{2}^{5} - 10 d x$

Step 8

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

$- (\frac{x^{3}}{3}]_{2}^{5}) + 7 (\frac{x^{2}}{2}]_{2}^{5}) + \int_{2}^{5} - 10 d x$

Step 9

Apply the constant rule.

$- (\frac{x^{3}}{3}]_{2}^{5}) + 7 (\frac{x^{2}}{2}]_{2}^{5}) + - 10 x]_{2}^{5}$

Step 10

Substitute and simplify.

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Step 10.1

Evaluate $\frac{x^{3}}{3}$ at $5$ and at $2$ .

$- ((\frac{5^{3}}{3}) - \frac{2^{3}}{3}) + 7 (\frac{x^{2}}{2}]_{2}^{5}) + - 10 x]_{2}^{5}$

Step 10.2

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

$- (\frac{5^{3}}{3} - \frac{2^{3}}{3}) + 7 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + - 10 x]_{2}^{5}$

Step 10.3

Evaluate $- 10 x$ at $5$ and at $2$ .

$- (\frac{5^{3}}{3} - \frac{2^{3}}{3}) + 7 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4

Simplify.

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Step 10.4.1

Raise $5$ to the power of $3$ .

$- (\frac{125}{3} - \frac{2^{3}}{3}) + 7 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.2

Raise $2$ to the power of $3$ .

$- (\frac{125}{3} - \frac{8}{3}) + 7 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.3

Combine the numerators over the common denominator.

$- \frac{125 - 8}{3} + 7 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.4

Subtract $8$ from $125$ .

$- \frac{117}{3} + 7 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.5

Cancel the common factor of $117$ and $3$ .

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Step 10.4.5.1

Factor $3$ out of $117$ .

$- \frac{3 \cdot 39}{3} + 7 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.5.2

Cancel the common factors.

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Step 10.4.5.2.1

Factor $3$ out of $3$ .

$- \frac{3 \cdot 39}{3 (1)} + 7 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.5.2.2

Cancel the common factor.

$- \frac{3 \cdot 39}{3 \cdot 1} + 7 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.5.2.3

Rewrite the expression.

$- \frac{39}{1} + 7 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.5.2.4

Divide $39$ by $1$ .

$- 1 \cdot 39 + 7 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.6

Multiply $- 1$ by $39$ .

$- 39 + 7 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.7

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

$- 39 + 7 (\frac{25}{2} - \frac{2^{2}}{2}) + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.8

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

$- 39 + 7 (\frac{25}{2} - \frac{4}{2}) + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.9

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

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Step 10.4.9.1

Factor $2$ out of $4$ .

$- 39 + 7 (\frac{25}{2} - \frac{2 \cdot 2}{2}) + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.9.2

Cancel the common factors.

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Step 10.4.9.2.1

Factor $2$ out of $2$ .

$- 39 + 7 (\frac{25}{2} - \frac{2 \cdot 2}{2 (1)}) + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.9.2.2

Cancel the common factor.

$- 39 + 7 (\frac{25}{2} - \frac{2 \cdot 2}{2 \cdot 1}) + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.9.2.3

Rewrite the expression.

$- 39 + 7 (\frac{25}{2} - \frac{2}{1}) + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.9.2.4

Divide $2$ by $1$ .

$- 39 + 7 (\frac{25}{2} - 1 \cdot 2) + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.10

Multiply $- 1$ by $2$ .

$- 39 + 7 (\frac{25}{2} - 2) + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.11

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

$- 39 + 7 (\frac{25}{2} - 2 \cdot \frac{2}{2}) + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.12

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

$- 39 + 7 (\frac{25}{2} + \frac{- 2 \cdot 2}{2}) + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.13

Combine the numerators over the common denominator.

$- 39 + 7 \frac{25 - 2 \cdot 2}{2} + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.14

Simplify the numerator.

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Step 10.4.14.1

Multiply $- 2$ by $2$ .

$- 39 + 7 \frac{25 - 4}{2} + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.14.2

Subtract $4$ from $25$ .

$- 39 + 7 (\frac{21}{2}) + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.15

Combine $7$ and $\frac{21}{2}$ .

$- 39 + \frac{7 \cdot 21}{2} + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.16

Multiply $7$ by $21$ .

$- 39 + \frac{147}{2} + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.17

To write $- 39$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 39 \cdot \frac{2}{2} + \frac{147}{2} + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.18

Combine $- 39$ and $\frac{2}{2}$ .

$\frac{- 39 \cdot 2}{2} + \frac{147}{2} + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.19

Combine the numerators over the common denominator.

$\frac{- 39 \cdot 2 + 147}{2} + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.20

Simplify the numerator.

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Step 10.4.20.1

Multiply $- 39$ by $2$ .

$\frac{- 78 + 147}{2} + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.20.2

Add $- 78$ and $147$ .

$\frac{69}{2} + (- 10 \cdot 5) + 10 \cdot 2$

Step 10.4.21

Multiply $- 10$ by $5$ .

$\frac{69}{2} - 50 + 10 \cdot 2$

Step 10.4.22

Multiply $10$ by $2$ .

$\frac{69}{2} - 50 + 20$

Step 10.4.23

Add $- 50$ and $20$ .

$\frac{69}{2} - 30$

Step 10.4.24

To write $- 30$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{69}{2} - 30 \cdot \frac{2}{2}$

Step 10.4.25

Combine $- 30$ and $\frac{2}{2}$ .

$\frac{69}{2} + \frac{- 30 \cdot 2}{2}$

Step 10.4.26

Combine the numerators over the common denominator.

$\frac{69 - 30 \cdot 2}{2}$

Step 10.4.27

Simplify the numerator.

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Step 10.4.27.1

Multiply $- 30$ by $2$ .

$\frac{69 - 60}{2}$

Step 10.4.27.2

Subtract $60$ from $69$ .

$\frac{9}{2}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{9}{2}$

Decimal Form:

$4.5$

Mixed Number Form:

$4 \frac{1}{2}$

Step 12