Calculus Examples

$\int_{- 1}^{4} (6 x + k) d x = 0$

Step 1

Remove parentheses.

$\int_{- 1}^{4} 6 x + k d x$

Step 2

Split the single integral into multiple integrals.

$\int_{- 1}^{4} 6 x d x + \int_{- 1}^{4} k d x$

Step 3

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

$6 \int_{- 1}^{4} x d x + \int_{- 1}^{4} k d x$

Step 4

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

$6 (\frac{1}{2} x^{2}]_{- 1}^{4}) + \int_{- 1}^{4} k d x$

Step 5

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

$6 (\frac{x^{2}}{2}]_{- 1}^{4}) + \int_{- 1}^{4} k d x$

Step 6

Apply the constant rule.

$6 (\frac{x^{2}}{2}]_{- 1}^{4}) + k x]_{- 1}^{4}$

Step 7

Substitute and simplify.

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Step 7.1

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

$6 ((\frac{4^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) + k x]_{- 1}^{4}$

Step 7.2

Evaluate $k x$ at $4$ and at $- 1$ .

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

Step 7.3

Simplify.

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Step 7.3.1

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

$6 (\frac{16}{2} - \frac{{(- 1)}^{2}}{2}) + k \cdot 4 - k \cdot - 1$

Step 7.3.2

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

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Step 7.3.2.1

Factor $2$ out of $16$ .

$6 (\frac{2 \cdot 8}{2} - \frac{{(- 1)}^{2}}{2}) + k \cdot 4 - k \cdot - 1$

Step 7.3.2.2

Cancel the common factors.

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Step 7.3.2.2.1

Factor $2$ out of $2$ .

$6 (\frac{2 \cdot 8}{2 (1)} - \frac{{(- 1)}^{2}}{2}) + k \cdot 4 - k \cdot - 1$

Step 7.3.2.2.2

Cancel the common factor.

$6 (\frac{2 \cdot 8}{2 \cdot 1} - \frac{{(- 1)}^{2}}{2}) + k \cdot 4 - k \cdot - 1$

Step 7.3.2.2.3

Rewrite the expression.

$6 (\frac{8}{1} - \frac{{(- 1)}^{2}}{2}) + k \cdot 4 - k \cdot - 1$

Step 7.3.2.2.4

Divide $8$ by $1$ .

$6 (8 - \frac{{(- 1)}^{2}}{2}) + k \cdot 4 - k \cdot - 1$

Step 7.3.3

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

$6 (8 - \frac{1}{2}) + k \cdot 4 - k \cdot - 1$

Step 7.3.4

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

$6 (8 \cdot \frac{2}{2} - \frac{1}{2}) + k \cdot 4 - k \cdot - 1$

Step 7.3.5

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

$6 (\frac{8 \cdot 2}{2} - \frac{1}{2}) + k \cdot 4 - k \cdot - 1$

Step 7.3.6

Combine the numerators over the common denominator.

$6 \frac{8 \cdot 2 - 1}{2} + k \cdot 4 - k \cdot - 1$

Step 7.3.7

Simplify the numerator.

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Step 7.3.7.1

Multiply $8$ by $2$ .

$6 \frac{16 - 1}{2} + k \cdot 4 - k \cdot - 1$

Step 7.3.7.2

Subtract $1$ from $16$ .

$6 (\frac{15}{2}) + k \cdot 4 - k \cdot - 1$

Step 7.3.8

Combine $6$ and $\frac{15}{2}$ .

$\frac{6 \cdot 15}{2} + k \cdot 4 - k \cdot - 1$

Step 7.3.9

Multiply $6$ by $15$ .

$\frac{90}{2} + k \cdot 4 - k \cdot - 1$

Step 7.3.10

Cancel the common factor of $90$ and $2$ .

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Step 7.3.10.1

Factor $2$ out of $90$ .

$\frac{2 \cdot 45}{2} + k \cdot 4 - k \cdot - 1$

Step 7.3.10.2

Cancel the common factors.

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Step 7.3.10.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 45}{2 (1)} + k \cdot 4 - k \cdot - 1$

Step 7.3.10.2.2

Cancel the common factor.

$\frac{2 \cdot 45}{2 \cdot 1} + k \cdot 4 - k \cdot - 1$

Step 7.3.10.2.3

Rewrite the expression.

$\frac{45}{1} + k \cdot 4 - k \cdot - 1$

Step 7.3.10.2.4

Divide $45$ by $1$ .

$45 + k \cdot 4 - k \cdot - 1$

Step 7.3.11

Move $4$ to the left of $k$ .

$45 + 4 \cdot k - k \cdot - 1$

Step 7.3.12

Multiply $- 1$ by $- 1$ .

$45 + 4 k + 1 k$

Step 7.3.13

Multiply $k$ by $1$ .

$45 + 4 k + k$

Step 7.3.14

Add $4 k$ and $k$ .

$45 + 5 k$

Step 8