Calculus Examples

$\int_{- 1}^{0} t^{\frac{1}{3}} - t^{\frac{2}{3}} d t$

Step 1

Split the single integral into multiple integrals.

$\int_{- 1}^{0} t^{\frac{1}{3}} d t + \int_{- 1}^{0} - t^{\frac{2}{3}} d t$

Step 2

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

$\frac{3}{4} t^{\frac{4}{3}}]_{- 1}^{0} + \int_{- 1}^{0} - t^{\frac{2}{3}} d t$

Step 3

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

$\frac{3}{4} t^{\frac{4}{3}}]_{- 1}^{0} - \int_{- 1}^{0} t^{\frac{2}{3}} d t$

Step 4

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

$\frac{3}{4} t^{\frac{4}{3}}]_{- 1}^{0} - (\frac{3}{5} t^{\frac{5}{3}}]_{- 1}^{0})$

Step 5

Simplify the answer.

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Step 5.1

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

$\frac{3}{4} t^{\frac{4}{3}}]_{- 1}^{0} - (\frac{3 t^{\frac{5}{3}}}{5}]_{- 1}^{0})$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $\frac{3}{4} t^{\frac{4}{3}}$ at $0$ and at $- 1$ .

$(\frac{3}{4} \cdot 0^{\frac{4}{3}}) - \frac{3}{4} {(- 1)}^{\frac{4}{3}} - (\frac{3 t^{\frac{5}{3}}}{5}]_{- 1}^{0})$

Step 5.2.2

Evaluate $\frac{3 t^{\frac{5}{3}}}{5}$ at $0$ and at $- 1$ .

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

Step 5.2.3

Simplify.

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Step 5.2.3.1

Rewrite $0$ as $0^{3}$ .

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

Step 5.2.3.2

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

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

Step 5.2.3.3

Cancel the common factor of $3$ .

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Step 5.2.3.3.1

Cancel the common factor.

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

Step 5.2.3.3.2

Rewrite the expression.

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

Step 5.2.3.4

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

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

Step 5.2.3.5

Multiply $\frac{3}{4}$ by $0$ .

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

Step 5.2.3.6

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

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

Step 5.2.3.7

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

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

Step 5.2.3.8

Cancel the common factor of $3$ .

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Step 5.2.3.8.1

Cancel the common factor.

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

Step 5.2.3.8.2

Rewrite the expression.

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

Step 5.2.3.9

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

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

Step 5.2.3.10

Multiply $- 1$ by $1$ .

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

Step 5.2.3.11

Subtract $\frac{3}{4}$ from $0$ .

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

Step 5.2.3.12

Rewrite $0$ as $0^{3}$ .

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

Step 5.2.3.13

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

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

Step 5.2.3.14

Cancel the common factor of $3$ .

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Step 5.2.3.14.1

Cancel the common factor.

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

Step 5.2.3.14.2

Rewrite the expression.

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

Step 5.2.3.15

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

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

Step 5.2.3.16

Multiply $3$ by $0$ .

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

Step 5.2.3.17

Cancel the common factor of $0$ and $5$ .

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Step 5.2.3.17.1

Factor $5$ out of $0$ .

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

Step 5.2.3.17.2

Cancel the common factors.

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Step 5.2.3.17.2.1

Factor $5$ out of $5$ .

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

Step 5.2.3.17.2.2

Cancel the common factor.

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

Step 5.2.3.17.2.3

Rewrite the expression.

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

Step 5.2.3.17.2.4

Divide $0$ by $1$ .

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

Step 5.2.3.18

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

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

Step 5.2.3.19

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

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

Step 5.2.3.20

Cancel the common factor of $3$ .

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Step 5.2.3.20.1

Cancel the common factor.

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

Step 5.2.3.20.2

Rewrite the expression.

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

Step 5.2.3.21

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

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

Step 5.2.3.22

Multiply $3$ by $- 1$ .

$- \frac{3}{4} - (0 - \frac{- 3}{5})$

Step 5.2.3.23

Move the negative in front of the fraction.

$- \frac{3}{4} - (0 - - \frac{3}{5})$

Step 5.2.3.24

Multiply $- 1$ by $- 1$ .

$- \frac{3}{4} - (0 + 1 (\frac{3}{5}))$

Step 5.2.3.25

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

$- \frac{3}{4} - (0 + \frac{3}{5})$

Step 5.2.3.26

Add $0$ and $\frac{3}{5}$ .

$- \frac{3}{4} - \frac{3}{5}$

Step 5.2.3.27

To write $- \frac{3}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{3}{4} \cdot \frac{5}{5} - \frac{3}{5}$

Step 5.2.3.28

To write $- \frac{3}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{3}{4} \cdot \frac{5}{5} - \frac{3}{5} \cdot \frac{4}{4}$

Step 5.2.3.29

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.3.29.1

Multiply $\frac{3}{4}$ by $\frac{5}{5}$ .

$- \frac{3 \cdot 5}{4 \cdot 5} - \frac{3}{5} \cdot \frac{4}{4}$

Step 5.2.3.29.2

Multiply $4$ by $5$ .

$- \frac{3 \cdot 5}{20} - \frac{3}{5} \cdot \frac{4}{4}$

Step 5.2.3.29.3

Multiply $\frac{3}{5}$ by $\frac{4}{4}$ .

$- \frac{3 \cdot 5}{20} - \frac{3 \cdot 4}{5 \cdot 4}$

Step 5.2.3.29.4

Multiply $5$ by $4$ .

$- \frac{3 \cdot 5}{20} - \frac{3 \cdot 4}{20}$

Step 5.2.3.30

Combine the numerators over the common denominator.

$\frac{- 3 \cdot 5 - 3 \cdot 4}{20}$

Step 5.2.3.31

Simplify the numerator.

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Step 5.2.3.31.1

Multiply $- 3$ by $5$ .

$\frac{- 15 - 3 \cdot 4}{20}$

Step 5.2.3.31.2

Multiply $- 3$ by $4$ .

$\frac{- 15 - 12}{20}$

Step 5.2.3.31.3

Subtract $12$ from $- 15$ .

$\frac{- 27}{20}$

Step 5.2.3.32

Move the negative in front of the fraction.

$- \frac{27}{20}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{27}{20}$

Decimal Form:

$- 1.35$

Mixed Number Form:

$- 1 \frac{7}{20}$

Step 7