Calculus Examples

$\int_{2}^{5} 4 - 2 x d x$

Step 1

Split the single integral into multiple integrals.

$\int_{2}^{5} 4 d x + \int_{2}^{5} - 2 x d x$

Step 2

Apply the constant rule.

$4 x]_{2}^{5} + \int_{2}^{5} - 2 x d x$

Step 3

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

$4 x]_{2}^{5} - 2 \int_{2}^{5} x d x$

Step 4

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

$4 x]_{2}^{5} - 2 (\frac{1}{2} x^{2}]_{2}^{5})$

Step 5

Simplify the answer.

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Step 5.1

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

$4 x]_{2}^{5} - 2 (\frac{x^{2}}{2}]_{2}^{5})$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $4 x$ at $5$ and at $2$ .

$(4 \cdot 5) - 4 \cdot 2 - 2 (\frac{x^{2}}{2}]_{2}^{5})$

Step 5.2.2

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

$4 \cdot 5 - 4 \cdot 2 - 2 (\frac{5^{2}}{2} - \frac{2^{2}}{2})$

Step 5.2.3

Simplify.

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Step 5.2.3.1

Multiply $4$ by $5$ .

$20 - 4 \cdot 2 - 2 (\frac{5^{2}}{2} - \frac{2^{2}}{2})$

Step 5.2.3.2

Multiply $- 4$ by $2$ .

$20 - 8 - 2 (\frac{5^{2}}{2} - \frac{2^{2}}{2})$

Step 5.2.3.3

Subtract $8$ from $20$ .

$12 - 2 (\frac{5^{2}}{2} - \frac{2^{2}}{2})$

Step 5.2.3.4

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

$12 - 2 (\frac{25}{2} - \frac{2^{2}}{2})$

Step 5.2.3.5

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

$12 - 2 (\frac{25}{2} - \frac{4}{2})$

Step 5.2.3.6

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

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Step 5.2.3.6.1

Factor $2$ out of $4$ .

$12 - 2 (\frac{25}{2} - \frac{2 \cdot 2}{2})$

Step 5.2.3.6.2

Cancel the common factors.

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Step 5.2.3.6.2.1

Factor $2$ out of $2$ .

$12 - 2 (\frac{25}{2} - \frac{2 \cdot 2}{2 (1)})$

Step 5.2.3.6.2.2

Cancel the common factor.

$12 - 2 (\frac{25}{2} - \frac{2 \cdot 2}{2 \cdot 1})$

Step 5.2.3.6.2.3

Rewrite the expression.

$12 - 2 (\frac{25}{2} - \frac{2}{1})$

Step 5.2.3.6.2.4

Divide $2$ by $1$ .

$12 - 2 (\frac{25}{2} - 1 \cdot 2)$

Step 5.2.3.7

Multiply $- 1$ by $2$ .

$12 - 2 (\frac{25}{2} - 2)$

Step 5.2.3.8

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

$12 - 2 (\frac{25}{2} - 2 \cdot \frac{2}{2})$

Step 5.2.3.9

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

$12 - 2 (\frac{25}{2} + \frac{- 2 \cdot 2}{2})$

Step 5.2.3.10

Combine the numerators over the common denominator.

$12 - 2 \frac{25 - 2 \cdot 2}{2}$

Step 5.2.3.11

Simplify the numerator.

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Step 5.2.3.11.1

Multiply $- 2$ by $2$ .

$12 - 2 \frac{25 - 4}{2}$

Step 5.2.3.11.2

Subtract $4$ from $25$ .

$12 - 2 (\frac{21}{2})$

Step 5.2.3.12

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

$12 + \frac{- 2 \cdot 21}{2}$

Step 5.2.3.13

Multiply $- 2$ by $21$ .

$12 + \frac{- 42}{2}$

Step 5.2.3.14

Cancel the common factor of $- 42$ and $2$ .

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Step 5.2.3.14.1

Factor $2$ out of $- 42$ .

$12 + \frac{2 \cdot - 21}{2}$

Step 5.2.3.14.2

Cancel the common factors.

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Step 5.2.3.14.2.1

Factor $2$ out of $2$ .

$12 + \frac{2 \cdot - 21}{2 (1)}$

Step 5.2.3.14.2.2

Cancel the common factor.

$12 + \frac{2 \cdot - 21}{2 \cdot 1}$

Step 5.2.3.14.2.3

Rewrite the expression.

$12 + \frac{- 21}{1}$

Step 5.2.3.14.2.4

Divide $- 21$ by $1$ .

$12 - 21$

Step 5.2.3.15

Subtract $21$ from $12$ .

$- 9$

Step 6