Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

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

Step 3

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

$2 (\frac{3}{4} u^{\frac{4}{3}}]_{- 1}^{1}) + \int_{- 1}^{1} - u^{\frac{2}{3}} d u$

Step 4

Combine $\frac{3}{4}$ and $u^{\frac{4}{3}}$ .

$2 (\frac{3 u^{\frac{4}{3}}}{4}]_{- 1}^{1}) + \int_{- 1}^{1} - u^{\frac{2}{3}} d u$

Step 5

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

$2 (\frac{3 u^{\frac{4}{3}}}{4}]_{- 1}^{1}) - \int_{- 1}^{1} u^{\frac{2}{3}} d u$

Step 6

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

$2 (\frac{3 u^{\frac{4}{3}}}{4}]_{- 1}^{1}) - (\frac{3}{5} u^{\frac{5}{3}}]_{- 1}^{1})$

Step 7

Simplify the answer.

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Step 7.1

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

$2 (\frac{3 u^{\frac{4}{3}}}{4}]_{- 1}^{1}) - (\frac{3 u^{\frac{5}{3}}}{5}]_{- 1}^{1})$

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $\frac{3 u^{\frac{4}{3}}}{4}$ at $1$ and at $- 1$ .

$2 ((\frac{3 \cdot 1^{\frac{4}{3}}}{4}) - \frac{3 {(- 1)}^{\frac{4}{3}}}{4}) - (\frac{3 u^{\frac{5}{3}}}{5}]_{- 1}^{1})$

Step 7.2.2

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

$2 (\frac{3 \cdot 1^{\frac{4}{3}}}{4} - \frac{3 {(- 1)}^{\frac{4}{3}}}{4}) - (\frac{3 \cdot 1^{\frac{5}{3}}}{5} - \frac{3 {(- 1)}^{\frac{5}{3}}}{5})$

Step 7.2.3

Simplify.

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Step 7.2.3.1

One to any power is one.

$2 (\frac{3 \cdot 1}{4} - \frac{3 {(- 1)}^{\frac{4}{3}}}{4}) - (\frac{3 \cdot 1^{\frac{5}{3}}}{5} - \frac{3 {(- 1)}^{\frac{5}{3}}}{5})$

Step 7.2.3.2

Multiply $3$ by $1$ .

$2 (\frac{3}{4} - \frac{3 {(- 1)}^{\frac{4}{3}}}{4}) - (\frac{3 \cdot 1^{\frac{5}{3}}}{5} - \frac{3 {(- 1)}^{\frac{5}{3}}}{5})$

Step 7.2.3.3

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

$2 (\frac{3}{4} - \frac{3 {({(- 1)}^{3})}^{\frac{4}{3}}}{4}) - (\frac{3 \cdot 1^{\frac{5}{3}}}{5} - \frac{3 {(- 1)}^{\frac{5}{3}}}{5})$

Step 7.2.3.4

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

$2 (\frac{3}{4} - \frac{3 {(- 1)}^{3 (\frac{4}{3})}}{4}) - (\frac{3 \cdot 1^{\frac{5}{3}}}{5} - \frac{3 {(- 1)}^{\frac{5}{3}}}{5})$

Step 7.2.3.5

Cancel the common factor of $3$ .

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Step 7.2.3.5.1

Cancel the common factor.

$2 (\frac{3}{4} - \frac{3 {(- 1)}^{3 (\frac{4}{3})}}{4}) - (\frac{3 \cdot 1^{\frac{5}{3}}}{5} - \frac{3 {(- 1)}^{\frac{5}{3}}}{5})$

Step 7.2.3.5.2

Rewrite the expression.

$2 (\frac{3}{4} - \frac{3 {(- 1)}^{4}}{4}) - (\frac{3 \cdot 1^{\frac{5}{3}}}{5} - \frac{3 {(- 1)}^{\frac{5}{3}}}{5})$

Step 7.2.3.6

Raise $- 1$ to the power of $4$ .

$2 (\frac{3}{4} - \frac{3 \cdot 1}{4}) - (\frac{3 \cdot 1^{\frac{5}{3}}}{5} - \frac{3 {(- 1)}^{\frac{5}{3}}}{5})$

Step 7.2.3.7

Multiply $3$ by $1$ .

$2 (\frac{3}{4} - \frac{3}{4}) - (\frac{3 \cdot 1^{\frac{5}{3}}}{5} - \frac{3 {(- 1)}^{\frac{5}{3}}}{5})$

Step 7.2.3.8

Combine the numerators over the common denominator.

$2 \frac{3 - 3}{4} - (\frac{3 \cdot 1^{\frac{5}{3}}}{5} - \frac{3 {(- 1)}^{\frac{5}{3}}}{5})$

Step 7.2.3.9

Subtract $3$ from $3$ .

$2 (\frac{0}{4}) - (\frac{3 \cdot 1^{\frac{5}{3}}}{5} - \frac{3 {(- 1)}^{\frac{5}{3}}}{5})$

Step 7.2.3.10

Cancel the common factor of $0$ and $4$ .

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Step 7.2.3.10.1

Factor $4$ out of $0$ .

$2 \frac{4 (0)}{4} - (\frac{3 \cdot 1^{\frac{5}{3}}}{5} - \frac{3 {(- 1)}^{\frac{5}{3}}}{5})$

Step 7.2.3.10.2

Cancel the common factors.

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Step 7.2.3.10.2.1

Factor $4$ out of $4$ .

$2 \frac{4 \cdot 0}{4 \cdot 1} - (\frac{3 \cdot 1^{\frac{5}{3}}}{5} - \frac{3 {(- 1)}^{\frac{5}{3}}}{5})$

Step 7.2.3.10.2.2

Cancel the common factor.

$2 \frac{4 \cdot 0}{4 \cdot 1} - (\frac{3 \cdot 1^{\frac{5}{3}}}{5} - \frac{3 {(- 1)}^{\frac{5}{3}}}{5})$

Step 7.2.3.10.2.3

Rewrite the expression.

$2 (\frac{0}{1}) - (\frac{3 \cdot 1^{\frac{5}{3}}}{5} - \frac{3 {(- 1)}^{\frac{5}{3}}}{5})$

Step 7.2.3.10.2.4

Divide $0$ by $1$ .

$2 \cdot 0 - (\frac{3 \cdot 1^{\frac{5}{3}}}{5} - \frac{3 {(- 1)}^{\frac{5}{3}}}{5})$

Step 7.2.3.11

Multiply $2$ by $0$ .

$0 - (\frac{3 \cdot 1^{\frac{5}{3}}}{5} - \frac{3 {(- 1)}^{\frac{5}{3}}}{5})$

Step 7.2.3.12

One to any power is one.

$0 - (\frac{3 \cdot 1}{5} - \frac{3 {(- 1)}^{\frac{5}{3}}}{5})$

Step 7.2.3.13

Multiply $3$ by $1$ .

$0 - (\frac{3}{5} - \frac{3 {(- 1)}^{\frac{5}{3}}}{5})$

Step 7.2.3.14

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

$0 - (\frac{3}{5} - \frac{3 {({(- 1)}^{3})}^{\frac{5}{3}}}{5})$

Step 7.2.3.15

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

$0 - (\frac{3}{5} - \frac{3 {(- 1)}^{3 (\frac{5}{3})}}{5})$

Step 7.2.3.16

Cancel the common factor of $3$ .

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Step 7.2.3.16.1

Cancel the common factor.

$0 - (\frac{3}{5} - \frac{3 {(- 1)}^{3 (\frac{5}{3})}}{5})$

Step 7.2.3.16.2

Rewrite the expression.

$0 - (\frac{3}{5} - \frac{3 {(- 1)}^{5}}{5})$

Step 7.2.3.17

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

$0 - (\frac{3}{5} - \frac{3 \cdot - 1}{5})$

Step 7.2.3.18

Multiply $3$ by $- 1$ .

$0 - (\frac{3}{5} - \frac{- 3}{5})$

Step 7.2.3.19

Move the negative in front of the fraction.

$0 - (\frac{3}{5} - - \frac{3}{5})$

Step 7.2.3.20

Multiply $- 1$ by $- 1$ .

$0 - (\frac{3}{5} + 1 (\frac{3}{5}))$

Step 7.2.3.21

Multiply $\frac{3}{5}$ by $1$ .

$0 - (\frac{3}{5} + \frac{3}{5})$

Step 7.2.3.22

Combine the numerators over the common denominator.

$0 - \frac{3 + 3}{5}$

Step 7.2.3.23

Add $3$ and $3$ .

$0 - \frac{6}{5}$

Step 7.2.3.24

Subtract $\frac{6}{5}$ from $0$ .

$- \frac{6}{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{6}{5}$

Decimal Form:

$- 1.2$

Mixed Number Form:

$- 1 \frac{1}{5}$

Step 9