Calculus Examples

$\int_{- 1}^{2} - (- 6 x - 10) d x$

Step 1

Multiply $- (- 6 x - 10)$ .

$\int_{- 1}^{2} - (- 6 x) - - 10 d x$

Step 2

Simplify.

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Step 2.1

Multiply $- 6$ by $- 1$ .

$\int_{- 1}^{2} 6 x - - 10 d x$

Step 2.2

Multiply $- 1$ by $- 10$ .

$\int_{- 1}^{2} 6 x + 10 d x$

Step 3

Split the single integral into multiple integrals.

$\int_{- 1}^{2} 6 x d x + \int_{- 1}^{2} 10 d x$

Step 4

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int_{- 1}^{2} x d x + \int_{- 1}^{2} 10 d x$

Step 5

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

$6 (\frac{1}{2} x^{2}]_{- 1}^{2}) + \int_{- 1}^{2} 10 d x$

Step 6

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

$6 (\frac{x^{2}}{2}]_{- 1}^{2}) + \int_{- 1}^{2} 10 d x$

Step 7

Apply the constant rule.

$6 (\frac{x^{2}}{2}]_{- 1}^{2}) + 10 x]_{- 1}^{2}$

Step 8

Substitute and simplify.

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Step 8.1

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

$6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) + 10 x]_{- 1}^{2}$

Step 8.2

Evaluate $10 x$ at $2$ and at $- 1$ .

$6 (\frac{2^{2}}{2} - \frac{{(- 1)}^{2}}{2}) + 10 \cdot 2 - 10 \cdot - 1$

Step 8.3

Simplify.

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Step 8.3.1

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

$6 (\frac{4}{2} - \frac{{(- 1)}^{2}}{2}) + 10 \cdot 2 - 10 \cdot - 1$

Step 8.3.2

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

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Step 8.3.2.1

Factor $2$ out of $4$ .

$6 (\frac{2 \cdot 2}{2} - \frac{{(- 1)}^{2}}{2}) + 10 \cdot 2 - 10 \cdot - 1$

Step 8.3.2.2

Cancel the common factors.

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Step 8.3.2.2.1

Factor $2$ out of $2$ .

$6 (\frac{2 \cdot 2}{2 (1)} - \frac{{(- 1)}^{2}}{2}) + 10 \cdot 2 - 10 \cdot - 1$

Step 8.3.2.2.2

Cancel the common factor.

$6 (\frac{2 \cdot 2}{2 \cdot 1} - \frac{{(- 1)}^{2}}{2}) + 10 \cdot 2 - 10 \cdot - 1$

Step 8.3.2.2.3

Rewrite the expression.

$6 (\frac{2}{1} - \frac{{(- 1)}^{2}}{2}) + 10 \cdot 2 - 10 \cdot - 1$

Step 8.3.2.2.4

Divide $2$ by $1$ .

$6 (2 - \frac{{(- 1)}^{2}}{2}) + 10 \cdot 2 - 10 \cdot - 1$

Step 8.3.3

Raise $- 1$ to the power of $2$ .

$6 (2 - \frac{1}{2}) + 10 \cdot 2 - 10 \cdot - 1$

Step 8.3.4

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

$6 (2 \cdot \frac{2}{2} - \frac{1}{2}) + 10 \cdot 2 - 10 \cdot - 1$

Step 8.3.5

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

$6 (\frac{2 \cdot 2}{2} - \frac{1}{2}) + 10 \cdot 2 - 10 \cdot - 1$

Step 8.3.6

Combine the numerators over the common denominator.

$6 \frac{2 \cdot 2 - 1}{2} + 10 \cdot 2 - 10 \cdot - 1$

Step 8.3.7

Simplify the numerator.

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Step 8.3.7.1

Multiply $2$ by $2$ .

$6 \frac{4 - 1}{2} + 10 \cdot 2 - 10 \cdot - 1$

Step 8.3.7.2

Subtract $1$ from $4$ .

$6 (\frac{3}{2}) + 10 \cdot 2 - 10 \cdot - 1$

Step 8.3.8

Combine $6$ and $\frac{3}{2}$ .

$\frac{6 \cdot 3}{2} + 10 \cdot 2 - 10 \cdot - 1$

Step 8.3.9

Multiply $6$ by $3$ .

$\frac{18}{2} + 10 \cdot 2 - 10 \cdot - 1$

Step 8.3.10

Cancel the common factor of $18$ and $2$ .

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Step 8.3.10.1

Factor $2$ out of $18$ .

$\frac{2 \cdot 9}{2} + 10 \cdot 2 - 10 \cdot - 1$

Step 8.3.10.2

Cancel the common factors.

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Step 8.3.10.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 9}{2 (1)} + 10 \cdot 2 - 10 \cdot - 1$

Step 8.3.10.2.2

Cancel the common factor.

$\frac{2 \cdot 9}{2 \cdot 1} + 10 \cdot 2 - 10 \cdot - 1$

Step 8.3.10.2.3

Rewrite the expression.

$\frac{9}{1} + 10 \cdot 2 - 10 \cdot - 1$

Step 8.3.10.2.4

Divide $9$ by $1$ .

$9 + 10 \cdot 2 - 10 \cdot - 1$

Step 8.3.11

Multiply $10$ by $2$ .

$9 + 20 - 10 \cdot - 1$

Step 8.3.12

Multiply $- 10$ by $- 1$ .

$9 + 20 + 10$

Step 8.3.13

Add $20$ and $10$ .

$9 + 30$

Step 8.3.14

Add $9$ and $30$ .

$39$

Step 9