Calculus Examples

$\int_{- 6}^{0} | x^{3} + 27 | d x$

Step 1

This integral could not be completed using u-substitution. Mathway will use another method.

Step 2

Split up the integral depending on where $x^{3} + 27$ is positive and negative.

$\int_{- 6}^{- 3} - x^{3} - 27 d x + \int_{- 3}^{0} x^{3} + 27 d x$

Step 3

Split the single integral into multiple integrals.

$\int_{- 6}^{- 3} - x^{3} d x + \int_{- 6}^{- 3} - 27 d x + \int_{- 3}^{0} x^{3} + 27 d x$

Step 4

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

$- \int_{- 6}^{- 3} x^{3} d x + \int_{- 6}^{- 3} - 27 d x + \int_{- 3}^{0} x^{3} + 27 d x$

Step 5

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$- (\frac{1}{4} x^{4}]_{- 6}^{- 3}) + \int_{- 6}^{- 3} - 27 d x + \int_{- 3}^{0} x^{3} + 27 d x$

Step 6

Combine $\frac{1}{4}$ and $x^{4}$ .

$- (\frac{x^{4}}{4}]_{- 6}^{- 3}) + \int_{- 6}^{- 3} - 27 d x + \int_{- 3}^{0} x^{3} + 27 d x$

Step 7

Apply the constant rule.

$- (\frac{x^{4}}{4}]_{- 6}^{- 3}) + - 27 x]_{- 6}^{- 3} + \int_{- 3}^{0} x^{3} + 27 d x$

Step 8

Split the single integral into multiple integrals.

$- (\frac{x^{4}}{4}]_{- 6}^{- 3}) + - 27 x]_{- 6}^{- 3} + \int_{- 3}^{0} x^{3} d x + \int_{- 3}^{0} 27 d x$

Step 9

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$- (\frac{x^{4}}{4}]_{- 6}^{- 3}) + - 27 x]_{- 6}^{- 3} + \frac{1}{4} x^{4}]_{- 3}^{0} + \int_{- 3}^{0} 27 d x$

Step 10

Apply the constant rule.

$- (\frac{x^{4}}{4}]_{- 6}^{- 3}) + - 27 x]_{- 6}^{- 3} + \frac{1}{4} x^{4}]_{- 3}^{0} + 27 x]_{- 3}^{0}$

Step 11

Simplify the answer.

Tap for more steps...

Step 11.1

Combine $\frac{1}{4} x^{4}]_{- 3}^{0}$ and $27 x]_{- 3}^{0}$ .

$- (\frac{x^{4}}{4}]_{- 6}^{- 3}) + - 27 x]_{- 6}^{- 3} + \frac{1}{4} x^{4} + 27 x]_{- 3}^{0}$

Step 11.2

Substitute and simplify.

Tap for more steps...

Step 11.2.1

Evaluate $\frac{x^{4}}{4}$ at $- 3$ and at $- 6$ .

$- ((\frac{{(- 3)}^{4}}{4}) - \frac{{(- 6)}^{4}}{4}) + - 27 x]_{- 6}^{- 3} + \frac{1}{4} x^{4} + 27 x]_{- 3}^{0}$

Step 11.2.2

Evaluate $- 27 x$ at $- 3$ and at $- 6$ .

$- (\frac{{(- 3)}^{4}}{4} - \frac{{(- 6)}^{4}}{4}) + - 27 \cdot - 3 + 27 \cdot - 6 + \frac{1}{4} x^{4} + 27 x]_{- 3}^{0}$

Step 11.2.3

Evaluate $\frac{1}{4} x^{4} + 27 x$ at $0$ and at $- 3$ .

$- (\frac{{(- 3)}^{4}}{4} - \frac{{(- 6)}^{4}}{4}) + - 27 \cdot - 3 + 27 \cdot - 6 + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4

Simplify.

Tap for more steps...

Step 11.2.4.1

Raise $- 3$ to the power of $4$ .

$- (\frac{81}{4} - \frac{{(- 6)}^{4}}{4}) - 27 \cdot - 3 + 27 \cdot - 6 + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.2

Raise $- 6$ to the power of $4$ .

$- (\frac{81}{4} - \frac{1296}{4}) - 27 \cdot - 3 + 27 \cdot - 6 + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.3

Cancel the common factor of $1296$ and $4$ .

Tap for more steps...

Step 11.2.4.3.1

Factor $4$ out of $1296$ .

$- (\frac{81}{4} - \frac{4 \cdot 324}{4}) - 27 \cdot - 3 + 27 \cdot - 6 + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.3.2

Cancel the common factors.

Tap for more steps...

Step 11.2.4.3.2.1

Factor $4$ out of $4$ .

$- (\frac{81}{4} - \frac{4 \cdot 324}{4 (1)}) - 27 \cdot - 3 + 27 \cdot - 6 + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.3.2.2

Cancel the common factor.

$- (\frac{81}{4} - \frac{4 \cdot 324}{4 \cdot 1}) - 27 \cdot - 3 + 27 \cdot - 6 + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.3.2.3

Rewrite the expression.

$- (\frac{81}{4} - \frac{324}{1}) - 27 \cdot - 3 + 27 \cdot - 6 + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.3.2.4

Divide $324$ by $1$ .

$- (\frac{81}{4} - 1 \cdot 324) - 27 \cdot - 3 + 27 \cdot - 6 + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.4

Multiply $- 1$ by $324$ .

$- (\frac{81}{4} - 324) - 27 \cdot - 3 + 27 \cdot - 6 + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.5

To write $- 324$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- (\frac{81}{4} - 324 \cdot \frac{4}{4}) - 27 \cdot - 3 + 27 \cdot - 6 + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.6

Combine $- 324$ and $\frac{4}{4}$ .

$- (\frac{81}{4} + \frac{- 324 \cdot 4}{4}) - 27 \cdot - 3 + 27 \cdot - 6 + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.7

Combine the numerators over the common denominator.

$- \frac{81 - 324 \cdot 4}{4} - 27 \cdot - 3 + 27 \cdot - 6 + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.8

Simplify the numerator.

Tap for more steps...

Step 11.2.4.8.1

Multiply $- 324$ by $4$ .

$- \frac{81 - 1296}{4} - 27 \cdot - 3 + 27 \cdot - 6 + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.8.2

Subtract $1296$ from $81$ .

$- \frac{- 1215}{4} - 27 \cdot - 3 + 27 \cdot - 6 + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.9

Move the negative in front of the fraction.

$- - \frac{1215}{4} - 27 \cdot - 3 + 27 \cdot - 6 + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.10

Multiply $- 1$ by $- 1$ .

$1 (\frac{1215}{4}) - 27 \cdot - 3 + 27 \cdot - 6 + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.11

Multiply $\frac{1215}{4}$ by $1$ .

$\frac{1215}{4} - 27 \cdot - 3 + 27 \cdot - 6 + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.12

Multiply $- 27$ by $- 3$ .

$\frac{1215}{4} + 81 + 27 \cdot - 6 + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.13

Multiply $27$ by $- 6$ .

$\frac{1215}{4} + 81 - 162 + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.14

Subtract $162$ from $81$ .

$\frac{1215}{4} - 81 + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.15

To write $- 81$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{1215}{4} - 81 \cdot \frac{4}{4} + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.16

Combine $- 81$ and $\frac{4}{4}$ .

$\frac{1215}{4} + \frac{- 81 \cdot 4}{4} + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.17

Combine the numerators over the common denominator.

$\frac{1215 - 81 \cdot 4}{4} + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.18

Simplify the numerator.

Tap for more steps...

Step 11.2.4.18.1

Multiply $- 81$ by $4$ .

$\frac{1215 - 324}{4} + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.18.2

Subtract $324$ from $1215$ .

$\frac{891}{4} + (\frac{1}{4} \cdot 0^{4} + 27 \cdot 0) - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.19

Raising $0$ to any positive power yields $0$ .

$\frac{891}{4} + \frac{1}{4} \cdot 0 + 27 \cdot 0 - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.20

Multiply $\frac{1}{4}$ by $0$ .

$\frac{891}{4} + 0 + 27 \cdot 0 - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.21

Multiply $27$ by $0$ .

$\frac{891}{4} + 0 + 0 - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.22

Add $0$ and $0$ .

$\frac{891}{4} + 0 - (\frac{1}{4} {(- 3)}^{4} + 27 \cdot - 3)$

Step 11.2.4.23

Raise $- 3$ to the power of $4$ .

$\frac{891}{4} + 0 - (\frac{1}{4} \cdot 81 + 27 \cdot - 3)$

Step 11.2.4.24

Combine $\frac{1}{4}$ and $81$ .

$\frac{891}{4} + 0 - (\frac{81}{4} + 27 \cdot - 3)$

Step 11.2.4.25

Multiply $27$ by $- 3$ .

$\frac{891}{4} + 0 - (\frac{81}{4} - 81)$

Step 11.2.4.26

To write $- 81$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{891}{4} + 0 - (\frac{81}{4} - 81 \cdot \frac{4}{4})$

Step 11.2.4.27

Combine $- 81$ and $\frac{4}{4}$ .

$\frac{891}{4} + 0 - (\frac{81}{4} + \frac{- 81 \cdot 4}{4})$

Step 11.2.4.28

Combine the numerators over the common denominator.

$\frac{891}{4} + 0 - \frac{81 - 81 \cdot 4}{4}$

Step 11.2.4.29

Simplify the numerator.

Tap for more steps...

Step 11.2.4.29.1

Multiply $- 81$ by $4$ .

$\frac{891}{4} + 0 - \frac{81 - 324}{4}$

Step 11.2.4.29.2

Subtract $324$ from $81$ .

$\frac{891}{4} + 0 - \frac{- 243}{4}$

Step 11.2.4.30

Move the negative in front of the fraction.

$\frac{891}{4} + 0 - - \frac{243}{4}$

Step 11.2.4.31

Multiply $- 1$ by $- 1$ .

$\frac{891}{4} + 0 + 1 (\frac{243}{4})$

Step 11.2.4.32

Multiply $\frac{243}{4}$ by $1$ .

$\frac{891}{4} + 0 + \frac{243}{4}$

Step 11.2.4.33

Add $0$ and $\frac{243}{4}$ .

$\frac{891}{4} + \frac{243}{4}$

Step 11.2.4.34

Combine the numerators over the common denominator.

$\frac{891 + 243}{4}$

Step 11.2.4.35

Add $891$ and $243$ .

$\frac{1134}{4}$

Step 11.2.4.36

Cancel the common factor of $1134$ and $4$ .

Tap for more steps...

Step 11.2.4.36.1

Factor $2$ out of $1134$ .

$\frac{2 (567)}{4}$

Step 11.2.4.36.2

Cancel the common factors.

Tap for more steps...

Step 11.2.4.36.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 567}{2 \cdot 2}$

Step 11.2.4.36.2.2

Cancel the common factor.

$\frac{2 \cdot 567}{2 \cdot 2}$

Step 11.2.4.36.2.3

Rewrite the expression.

$\frac{567}{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{567}{2}$

Decimal Form:

$283.5$

Mixed Number Form:

$283 \frac{1}{2}$

Step 13