Calculus Examples

$\int_{1}^{4} 10 - 4 x d x$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{4} 10 d x + \int_{1}^{4} - 4 x d x$

Step 2

Apply the constant rule.

$10 x]_{1}^{4} + \int_{1}^{4} - 4 x d x$

Step 3

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$10 x]_{1}^{4} - 4 \int_{1}^{4} x d x$

Step 4

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

$10 x]_{1}^{4} - 4 (\frac{1}{2} x^{2}]_{1}^{4})$

Step 5

Simplify the answer.

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Step 5.1

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

$10 x]_{1}^{4} - 4 (\frac{x^{2}}{2}]_{1}^{4})$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $10 x$ at $4$ and at $1$ .

$(10 \cdot 4) - 10 \cdot 1 - 4 (\frac{x^{2}}{2}]_{1}^{4})$

Step 5.2.2

Evaluate $\frac{x^{2}}{2}$ at $4$ and at $1$ .

$10 \cdot 4 - 10 \cdot 1 - 4 (\frac{4^{2}}{2} - \frac{1^{2}}{2})$

Step 5.2.3

Simplify.

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Step 5.2.3.1

Multiply $10$ by $4$ .

$40 - 10 \cdot 1 - 4 (\frac{4^{2}}{2} - \frac{1^{2}}{2})$

Step 5.2.3.2

Multiply $- 10$ by $1$ .

$40 - 10 - 4 (\frac{4^{2}}{2} - \frac{1^{2}}{2})$

Step 5.2.3.3

Subtract $10$ from $40$ .

$30 - 4 (\frac{4^{2}}{2} - \frac{1^{2}}{2})$

Step 5.2.3.4

Raise $4$ to the power of $2$ .

$30 - 4 (\frac{16}{2} - \frac{1^{2}}{2})$

Step 5.2.3.5

Cancel the common factor of $16$ and $2$ .

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Step 5.2.3.5.1

Factor $2$ out of $16$ .

$30 - 4 (\frac{2 \cdot 8}{2} - \frac{1^{2}}{2})$

Step 5.2.3.5.2

Cancel the common factors.

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Step 5.2.3.5.2.1

Factor $2$ out of $2$ .

$30 - 4 (\frac{2 \cdot 8}{2 (1)} - \frac{1^{2}}{2})$

Step 5.2.3.5.2.2

Cancel the common factor.

$30 - 4 (\frac{2 \cdot 8}{2 \cdot 1} - \frac{1^{2}}{2})$

Step 5.2.3.5.2.3

Rewrite the expression.

$30 - 4 (\frac{8}{1} - \frac{1^{2}}{2})$

Step 5.2.3.5.2.4

Divide $8$ by $1$ .

$30 - 4 (8 - \frac{1^{2}}{2})$

Step 5.2.3.6

One to any power is one.

$30 - 4 (8 - \frac{1}{2})$

Step 5.2.3.7

To write $8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$30 - 4 (8 \cdot \frac{2}{2} - \frac{1}{2})$

Step 5.2.3.8

Combine $8$ and $\frac{2}{2}$ .

$30 - 4 (\frac{8 \cdot 2}{2} - \frac{1}{2})$

Step 5.2.3.9

Combine the numerators over the common denominator.

$30 - 4 \frac{8 \cdot 2 - 1}{2}$

Step 5.2.3.10

Simplify the numerator.

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Step 5.2.3.10.1

Multiply $8$ by $2$ .

$30 - 4 \frac{16 - 1}{2}$

Step 5.2.3.10.2

Subtract $1$ from $16$ .

$30 - 4 (\frac{15}{2})$

Step 5.2.3.11

Combine $- 4$ and $\frac{15}{2}$ .

$30 + \frac{- 4 \cdot 15}{2}$

Step 5.2.3.12

Multiply $- 4$ by $15$ .

$30 + \frac{- 60}{2}$

Step 5.2.3.13

Cancel the common factor of $- 60$ and $2$ .

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Step 5.2.3.13.1

Factor $2$ out of $- 60$ .

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

Step 5.2.3.13.2

Cancel the common factors.

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Step 5.2.3.13.2.1

Factor $2$ out of $2$ .

$30 + \frac{2 \cdot - 30}{2 (1)}$

Step 5.2.3.13.2.2

Cancel the common factor.

$30 + \frac{2 \cdot - 30}{2 \cdot 1}$

Step 5.2.3.13.2.3

Rewrite the expression.

$30 + \frac{- 30}{1}$

Step 5.2.3.13.2.4

Divide $- 30$ by $1$ .

$30 - 30$

Step 5.2.3.14

Subtract $30$ from $30$ .

$0$

Step 6