Calculus Examples

$\int_{- 8}^{- 1} \frac{x - x^{2}}{2 \sqrt[3]{x}} d x$

Step 1

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{- 8}^{- 1} \frac{x - x^{2}}{\sqrt[3]{x}} d x$

Step 2

Simplify the expression.

Tap for more steps...

Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$\frac{1}{2} \int_{- 8}^{- 1} \frac{x - x^{2}}{x^{\frac{1}{3}}} d x$

Step 2.2

Simplify.

Tap for more steps...

Step 2.2.1

Factor $x$ out of $x - x^{2}$ .

Tap for more steps...

Step 2.2.1.1

Raise $x$ to the power of $1$ .

$\frac{1}{2} \int_{- 8}^{- 1} \frac{x^{1} - x^{2}}{x^{\frac{1}{3}}} d x$

Step 2.2.1.2

Factor $x$ out of $x^{1}$ .

$\frac{1}{2} \int_{- 8}^{- 1} \frac{x \cdot 1 - x^{2}}{x^{\frac{1}{3}}} d x$

Step 2.2.1.3

Factor $x$ out of $- x^{2}$ .

$\frac{1}{2} \int_{- 8}^{- 1} \frac{x \cdot 1 + x (- x)}{x^{\frac{1}{3}}} d x$

Step 2.2.1.4

Factor $x$ out of $x \cdot 1 + x (- x)$ .

$\frac{1}{2} \int_{- 8}^{- 1} \frac{x (1 - x)}{x^{\frac{1}{3}}} d x$

Step 2.2.2

Move $x^{\frac{1}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{1}{2} \int_{- 8}^{- 1} x (1 - x) x^{- \frac{1}{3}} d x$

Step 2.2.3

Multiply $x$ by $x^{- \frac{1}{3}}$ by adding the exponents.

Tap for more steps...

Step 2.2.3.1

Move $x^{- \frac{1}{3}}$ .

$\frac{1}{2} \int_{- 8}^{- 1} x^{- \frac{1}{3}} x (1 - x) d x$

Step 2.2.3.2

Multiply $x^{- \frac{1}{3}}$ by $x$ .

Tap for more steps...

Step 2.2.3.2.1

Raise $x$ to the power of $1$ .

$\frac{1}{2} \int_{- 8}^{- 1} x^{- \frac{1}{3}} x^{1} (1 - x) d x$

Step 2.2.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{2} \int_{- 8}^{- 1} x^{- \frac{1}{3} + 1} (1 - x) d x$

Step 2.2.3.3

Write $1$ as a fraction with a common denominator.

$\frac{1}{2} \int_{- 8}^{- 1} x^{- \frac{1}{3} + \frac{3}{3}} (1 - x) d x$

Step 2.2.3.4

Combine the numerators over the common denominator.

$\frac{1}{2} \int_{- 8}^{- 1} x^{\frac{- 1 + 3}{3}} (1 - x) d x$

Step 2.2.3.5

Add $- 1$ and $3$ .

$\frac{1}{2} \int_{- 8}^{- 1} x^{\frac{2}{3}} (1 - x) d x$

Step 3

Expand $x^{\frac{2}{3}} (1 - x)$ .

Tap for more steps...

Step 3.1

Apply the distributive property.

$\frac{1}{2} \int_{- 8}^{- 1} x^{\frac{2}{3}} \cdot 1 + x^{\frac{2}{3}} (- x) d x$

Step 3.2

Reorder $x^{\frac{2}{3}}$ and $1$ .

$\frac{1}{2} \int_{- 8}^{- 1} 1 \cdot x^{\frac{2}{3}} + x^{\frac{2}{3}} (- x) d x$

Step 3.3

Reorder $x^{\frac{2}{3}}$ and $- 1$ .

$\frac{1}{2} \int_{- 8}^{- 1} 1 \cdot x^{\frac{2}{3}} - 1 \cdot x^{\frac{2}{3}} x d x$

Step 3.4

Multiply $x^{\frac{2}{3}}$ by $1$ .

$\frac{1}{2} \int_{- 8}^{- 1} x^{\frac{2}{3}} - 1 x^{\frac{2}{3}} x d x$

Step 3.5

Factor out negative.

$\frac{1}{2} \int_{- 8}^{- 1} x^{\frac{2}{3}} - (x^{\frac{2}{3}} x) d x$

Step 3.6

Raise $x$ to the power of $1$ .

$\frac{1}{2} \int_{- 8}^{- 1} x^{\frac{2}{3}} - (x^{\frac{2}{3}} x^{1}) d x$

Step 3.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{2} \int_{- 8}^{- 1} x^{\frac{2}{3}} - x^{\frac{2}{3} + 1} d x$

Step 3.8

Write $1$ as a fraction with a common denominator.

$\frac{1}{2} \int_{- 8}^{- 1} x^{\frac{2}{3}} - x^{\frac{2}{3} + \frac{3}{3}} d x$

Step 3.9

Combine the numerators over the common denominator.

$\frac{1}{2} \int_{- 8}^{- 1} x^{\frac{2}{3}} - x^{\frac{2 + 3}{3}} d x$

Step 3.10

Add $2$ and $3$ .

$\frac{1}{2} \int_{- 8}^{- 1} x^{\frac{2}{3}} - x^{\frac{5}{3}} d x$

Step 4

Split the single integral into multiple integrals.

$\frac{1}{2} (\int_{- 8}^{- 1} x^{\frac{2}{3}} d x + \int_{- 8}^{- 1} - x^{\frac{5}{3}} d x)$

Step 5

By the Power Rule, the integral of $x^{\frac{2}{3}}$ with respect to $x$ is $\frac{3}{5} x^{\frac{5}{3}}$ .

$\frac{1}{2} (\frac{3}{5} x^{\frac{5}{3}}]_{- 8}^{- 1} + \int_{- 8}^{- 1} - x^{\frac{5}{3}} d x)$

Step 6

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

$\frac{1}{2} (\frac{3}{5} x^{\frac{5}{3}}]_{- 8}^{- 1} - \int_{- 8}^{- 1} x^{\frac{5}{3}} d x)$

Step 7

By the Power Rule, the integral of $x^{\frac{5}{3}}$ with respect to $x$ is $\frac{3}{8} x^{\frac{8}{3}}$ .

$\frac{1}{2} (\frac{3}{5} x^{\frac{5}{3}}]_{- 8}^{- 1} - (\frac{3}{8} x^{\frac{8}{3}}]_{- 8}^{- 1}))$

Step 8

Simplify the answer.

Tap for more steps...

Step 8.1

Simplify.

Tap for more steps...

Step 8.1.1

Combine $\frac{3}{8}$ and $x^{\frac{8}{3}}$ .

$\frac{1}{2} (\frac{3}{5} x^{\frac{5}{3}}]_{- 8}^{- 1} - (\frac{3 x^{\frac{8}{3}}}{8}]_{- 8}^{- 1}))$

Step 8.1.2

Combine $\frac{3}{5}$ and $x^{\frac{5}{3}}$ .

$\frac{1}{2} (\frac{3 x^{\frac{5}{3}}}{5}]_{- 8}^{- 1} - (\frac{3 x^{\frac{8}{3}}}{8}]_{- 8}^{- 1}))$

Step 8.2

Substitute and simplify.

Tap for more steps...

Step 8.2.1

Evaluate $\frac{3 x^{\frac{5}{3}}}{5}$ at $- 1$ and at $- 8$ .

$\frac{1}{2} ((\frac{3 {(- 1)}^{\frac{5}{3}}}{5}) - \frac{3 {(- 8)}^{\frac{5}{3}}}{5} - (\frac{3 x^{\frac{8}{3}}}{8}]_{- 8}^{- 1}))$

Step 8.2.2

Evaluate $\frac{3 x^{\frac{8}{3}}}{8}$ at $- 1$ and at $- 8$ .

$\frac{1}{2} ((\frac{3 {(- 1)}^{\frac{5}{3}}}{5}) - \frac{3 {(- 8)}^{\frac{5}{3}}}{5} - ((\frac{3 {(- 1)}^{\frac{8}{3}}}{8}) - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3

Simplify.

Tap for more steps...

Step 8.2.3.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$\frac{1}{2} (\frac{3 {({(- 1)}^{3})}^{\frac{5}{3}}}{5} - \frac{3 {(- 8)}^{\frac{5}{3}}}{5} - ((\frac{3 {(- 1)}^{\frac{8}{3}}}{8}) - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{2} (\frac{3 {(- 1)}^{3 (\frac{5}{3})}}{5} - \frac{3 {(- 8)}^{\frac{5}{3}}}{5} - ((\frac{3 {(- 1)}^{\frac{8}{3}}}{8}) - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 8.2.3.3.1

Cancel the common factor.

$\frac{1}{2} (\frac{3 {(- 1)}^{3 (\frac{5}{3})}}{5} - \frac{3 {(- 8)}^{\frac{5}{3}}}{5} - ((\frac{3 {(- 1)}^{\frac{8}{3}}}{8}) - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.3.2

Rewrite the expression.

$\frac{1}{2} (\frac{3 {(- 1)}^{5}}{5} - \frac{3 {(- 8)}^{\frac{5}{3}}}{5} - ((\frac{3 {(- 1)}^{\frac{8}{3}}}{8}) - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.4

Raise $- 1$ to the power of $5$ .

$\frac{1}{2} (\frac{3 \cdot - 1}{5} - \frac{3 {(- 8)}^{\frac{5}{3}}}{5} - ((\frac{3 {(- 1)}^{\frac{8}{3}}}{8}) - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.5

Multiply $3$ by $- 1$ .

$\frac{1}{2} (\frac{- 3}{5} - \frac{3 {(- 8)}^{\frac{5}{3}}}{5} - ((\frac{3 {(- 1)}^{\frac{8}{3}}}{8}) - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.6

Move the negative in front of the fraction.

$\frac{1}{2} (- \frac{3}{5} - \frac{3 {(- 8)}^{\frac{5}{3}}}{5} - ((\frac{3 {(- 1)}^{\frac{8}{3}}}{8}) - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.7

Rewrite $- 8$ as ${(- 2)}^{3}$ .

$\frac{1}{2} (- \frac{3}{5} - \frac{3 {({(- 2)}^{3})}^{\frac{5}{3}}}{5} - ((\frac{3 {(- 1)}^{\frac{8}{3}}}{8}) - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.8

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{2} (- \frac{3}{5} - \frac{3 {(- 2)}^{3 (\frac{5}{3})}}{5} - ((\frac{3 {(- 1)}^{\frac{8}{3}}}{8}) - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.9

Cancel the common factor of $3$ .

Tap for more steps...

Step 8.2.3.9.1

Cancel the common factor.

$\frac{1}{2} (- \frac{3}{5} - \frac{3 {(- 2)}^{3 (\frac{5}{3})}}{5} - ((\frac{3 {(- 1)}^{\frac{8}{3}}}{8}) - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.9.2

Rewrite the expression.

$\frac{1}{2} (- \frac{3}{5} - \frac{3 {(- 2)}^{5}}{5} - ((\frac{3 {(- 1)}^{\frac{8}{3}}}{8}) - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.10

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

$\frac{1}{2} (- \frac{3}{5} - \frac{3 \cdot - 32}{5} - ((\frac{3 {(- 1)}^{\frac{8}{3}}}{8}) - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.11

Multiply $3$ by $- 32$ .

$\frac{1}{2} (- \frac{3}{5} - \frac{- 96}{5} - ((\frac{3 {(- 1)}^{\frac{8}{3}}}{8}) - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.12

Move the negative in front of the fraction.

$\frac{1}{2} (- \frac{3}{5} - - \frac{96}{5} - ((\frac{3 {(- 1)}^{\frac{8}{3}}}{8}) - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.13

Multiply $- 1$ by $- 1$ .

$\frac{1}{2} (- \frac{3}{5} + 1 (\frac{96}{5}) - ((\frac{3 {(- 1)}^{\frac{8}{3}}}{8}) - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.14

Multiply $\frac{96}{5}$ by $1$ .

$\frac{1}{2} (- \frac{3}{5} + \frac{96}{5} - ((\frac{3 {(- 1)}^{\frac{8}{3}}}{8}) - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.15

Combine the numerators over the common denominator.

$\frac{1}{2} (\frac{- 3 + 96}{5} - ((\frac{3 {(- 1)}^{\frac{8}{3}}}{8}) - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.16

Add $- 3$ and $96$ .

$\frac{1}{2} (\frac{93}{5} - ((\frac{3 {(- 1)}^{\frac{8}{3}}}{8}) - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.17

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$\frac{1}{2} (\frac{93}{5} - (\frac{3 {({(- 1)}^{3})}^{\frac{8}{3}}}{8} - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.18

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{2} (\frac{93}{5} - (\frac{3 {(- 1)}^{3 (\frac{8}{3})}}{8} - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.19

Cancel the common factor of $3$ .

Tap for more steps...

Step 8.2.3.19.1

Cancel the common factor.

$\frac{1}{2} (\frac{93}{5} - (\frac{3 {(- 1)}^{3 (\frac{8}{3})}}{8} - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.19.2

Rewrite the expression.

$\frac{1}{2} (\frac{93}{5} - (\frac{3 {(- 1)}^{8}}{8} - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.20

Raise $- 1$ to the power of $8$ .

$\frac{1}{2} (\frac{93}{5} - (\frac{3 \cdot 1}{8} - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.21

Multiply $3$ by $1$ .

$\frac{1}{2} (\frac{93}{5} - (\frac{3}{8} - \frac{3 {(- 8)}^{\frac{8}{3}}}{8}))$

Step 8.2.3.22

Rewrite $- 8$ as ${(- 2)}^{3}$ .

$\frac{1}{2} (\frac{93}{5} - (\frac{3}{8} - \frac{3 {({(- 2)}^{3})}^{\frac{8}{3}}}{8}))$

Step 8.2.3.23

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{2} (\frac{93}{5} - (\frac{3}{8} - \frac{3 {(- 2)}^{3 (\frac{8}{3})}}{8}))$

Step 8.2.3.24

Cancel the common factor of $3$ .

Tap for more steps...

Step 8.2.3.24.1

Cancel the common factor.

$\frac{1}{2} (\frac{93}{5} - (\frac{3}{8} - \frac{3 {(- 2)}^{3 (\frac{8}{3})}}{8}))$

Step 8.2.3.24.2

Rewrite the expression.

$\frac{1}{2} (\frac{93}{5} - (\frac{3}{8} - \frac{3 {(- 2)}^{8}}{8}))$

Step 8.2.3.25

Raise $- 2$ to the power of $8$ .

$\frac{1}{2} (\frac{93}{5} - (\frac{3}{8} - \frac{3 \cdot 256}{8}))$

Step 8.2.3.26

Multiply $3$ by $256$ .

$\frac{1}{2} (\frac{93}{5} - (\frac{3}{8} - \frac{768}{8}))$

Step 8.2.3.27

Cancel the common factor of $768$ and $8$ .

Tap for more steps...

Step 8.2.3.27.1

Factor $8$ out of $768$ .

$\frac{1}{2} (\frac{93}{5} - (\frac{3}{8} - \frac{8 \cdot 96}{8}))$

Step 8.2.3.27.2

Cancel the common factors.

Tap for more steps...

Step 8.2.3.27.2.1

Factor $8$ out of $8$ .

$\frac{1}{2} (\frac{93}{5} - (\frac{3}{8} - \frac{8 \cdot 96}{8 (1)}))$

Step 8.2.3.27.2.2

Cancel the common factor.

$\frac{1}{2} (\frac{93}{5} - (\frac{3}{8} - \frac{8 \cdot 96}{8 \cdot 1}))$

Step 8.2.3.27.2.3

Rewrite the expression.

$\frac{1}{2} (\frac{93}{5} - (\frac{3}{8} - \frac{96}{1}))$

Step 8.2.3.27.2.4

Divide $96$ by $1$ .

$\frac{1}{2} (\frac{93}{5} - (\frac{3}{8} - 1 \cdot 96))$

Step 8.2.3.28

Multiply $- 1$ by $96$ .

$\frac{1}{2} (\frac{93}{5} - (\frac{3}{8} - 96))$

Step 8.2.3.29

To write $- 96$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{1}{2} (\frac{93}{5} - (\frac{3}{8} - 96 \cdot \frac{8}{8}))$

Step 8.2.3.30

Combine $- 96$ and $\frac{8}{8}$ .

$\frac{1}{2} (\frac{93}{5} - (\frac{3}{8} + \frac{- 96 \cdot 8}{8}))$

Step 8.2.3.31

Combine the numerators over the common denominator.

$\frac{1}{2} (\frac{93}{5} - \frac{3 - 96 \cdot 8}{8})$

Step 8.2.3.32

Simplify the numerator.

Tap for more steps...

Step 8.2.3.32.1

Multiply $- 96$ by $8$ .

$\frac{1}{2} (\frac{93}{5} - \frac{3 - 768}{8})$

Step 8.2.3.32.2

Subtract $768$ from $3$ .

$\frac{1}{2} (\frac{93}{5} - \frac{- 765}{8})$

Step 8.2.3.33

Move the negative in front of the fraction.

$\frac{1}{2} (\frac{93}{5} - - \frac{765}{8})$

Step 8.2.3.34

Multiply $- 1$ by $- 1$ .

$\frac{1}{2} (\frac{93}{5} + 1 (\frac{765}{8}))$

Step 8.2.3.35

Multiply $\frac{765}{8}$ by $1$ .

$\frac{1}{2} (\frac{93}{5} + \frac{765}{8})$

Step 8.2.3.36

To write $\frac{93}{5}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{1}{2} (\frac{93}{5} \cdot \frac{8}{8} + \frac{765}{8})$

Step 8.2.3.37

To write $\frac{765}{8}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{1}{2} (\frac{93}{5} \cdot \frac{8}{8} + \frac{765}{8} \cdot \frac{5}{5})$

Step 8.2.3.38

Write each expression with a common denominator of $40$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 8.2.3.38.1

Multiply $\frac{93}{5}$ by $\frac{8}{8}$ .

$\frac{1}{2} (\frac{93 \cdot 8}{5 \cdot 8} + \frac{765}{8} \cdot \frac{5}{5})$

Step 8.2.3.38.2

Multiply $5$ by $8$ .

$\frac{1}{2} (\frac{93 \cdot 8}{40} + \frac{765}{8} \cdot \frac{5}{5})$

Step 8.2.3.38.3

Multiply $\frac{765}{8}$ by $\frac{5}{5}$ .

$\frac{1}{2} (\frac{93 \cdot 8}{40} + \frac{765 \cdot 5}{8 \cdot 5})$

Step 8.2.3.38.4

Multiply $8$ by $5$ .

$\frac{1}{2} (\frac{93 \cdot 8}{40} + \frac{765 \cdot 5}{40})$

Step 8.2.3.39

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot \frac{93 \cdot 8 + 765 \cdot 5}{40}$

Step 8.2.3.40

Simplify the numerator.

Tap for more steps...

Step 8.2.3.40.1

Multiply $93$ by $8$ .

$\frac{1}{2} \cdot \frac{744 + 765 \cdot 5}{40}$

Step 8.2.3.40.2

Multiply $765$ by $5$ .

$\frac{1}{2} \cdot \frac{744 + 3825}{40}$

Step 8.2.3.40.3

Add $744$ and $3825$ .

$\frac{1}{2} \cdot \frac{4569}{40}$

Step 8.2.3.41

Multiply $\frac{1}{2}$ by $\frac{4569}{40}$ .

$\frac{4569}{2 \cdot 40}$

Step 8.2.3.42

Multiply $2$ by $40$ .

$\frac{4569}{80}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{4569}{80}$

Decimal Form:

$57.1125$

Mixed Number Form:

$57 \frac{9}{80}$

Step 10