Calculus Examples

$\int_{0}^{2} (2 - t) \sqrt{t} d t$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Factor out negative.

$\int_{0}^{2} 2 t^{\frac{1}{2}} - (t \cdot t^{\frac{1}{2}}) d t$

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Add $2$ and $1$ .

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

By the Power Rule, the integral of $t^{\frac{1}{2}}$ with respect to $t$ is $\frac{2}{3} t^{\frac{3}{2}}$ .

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

Step 6

Combine $\frac{2}{3}$ and $t^{\frac{3}{2}}$ .

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

Step 7

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

$2 (\frac{2 t^{\frac{3}{2}}}{3}]_{0}^{2}) - \int_{0}^{2} t^{\frac{3}{2}} d t$

Step 8

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

$2 (\frac{2 t^{\frac{3}{2}}}{3}]_{0}^{2}) - (\frac{2}{5} t^{\frac{5}{2}}]_{0}^{2})$

Step 9

Simplify the answer.

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Step 9.1

Combine $\frac{2}{5}$ and $t^{\frac{5}{2}}$ .

$2 (\frac{2 t^{\frac{3}{2}}}{3}]_{0}^{2}) - (\frac{2 t^{\frac{5}{2}}}{5}]_{0}^{2})$

Step 9.2

Substitute and simplify.

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Step 9.2.1

Evaluate $\frac{2 t^{\frac{3}{2}}}{3}$ at $2$ and at $0$ .

$2 ((\frac{2 \cdot 2^{\frac{3}{2}}}{3}) - \frac{2 \cdot 0^{\frac{3}{2}}}{3}) - (\frac{2 t^{\frac{5}{2}}}{5}]_{0}^{2})$

Step 9.2.2

Evaluate $\frac{2 t^{\frac{5}{2}}}{5}$ at $2$ and at $0$ .

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

Step 9.2.3

Simplify.

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Step 9.2.3.1

Multiply $2$ by $2^{\frac{3}{2}}$ by adding the exponents.

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Step 9.2.3.1.1

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

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Step 9.2.3.1.1.1

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

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

Step 9.2.3.1.1.2

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

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

Step 9.2.3.1.2

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

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

Step 9.2.3.1.3

Combine the numerators over the common denominator.

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

Step 9.2.3.1.4

Add $2$ and $3$ .

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

Step 9.2.3.2

Rewrite $0$ as $0^{2}$ .

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

Step 9.2.3.3

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

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

Step 9.2.3.4

Cancel the common factor of $2$ .

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Step 9.2.3.4.1

Cancel the common factor.

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

Step 9.2.3.4.2

Rewrite the expression.

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

Step 9.2.3.5

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

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

Step 9.2.3.6

Multiply $2$ by $0$ .

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

Step 9.2.3.7

Cancel the common factor of $0$ and $3$ .

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Step 9.2.3.7.1

Factor $3$ out of $0$ .

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

Step 9.2.3.7.2

Cancel the common factors.

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Step 9.2.3.7.2.1

Factor $3$ out of $3$ .

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

Step 9.2.3.7.2.2

Cancel the common factor.

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

Step 9.2.3.7.2.3

Rewrite the expression.

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

Step 9.2.3.7.2.4

Divide $0$ by $1$ .

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

Step 9.2.3.8

Multiply $- 1$ by $0$ .

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

Step 9.2.3.9

Add $\frac{2^{\frac{5}{2}}}{3}$ and $0$ .

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

Step 9.2.3.10

Combine $2$ and $\frac{2^{\frac{5}{2}}}{3}$ .

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

Step 9.2.3.11

Multiply $2$ by $2^{\frac{5}{2}}$ by adding the exponents.

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Step 9.2.3.11.1

Multiply $2$ by $2^{\frac{5}{2}}$ .

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Step 9.2.3.11.1.1

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

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

Step 9.2.3.11.1.2

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

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

Step 9.2.3.11.2

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

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

Step 9.2.3.11.3

Combine the numerators over the common denominator.

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

Step 9.2.3.11.4

Add $2$ and $5$ .

$\frac{2^{\frac{7}{2}}}{3} - (\frac{2 \cdot 2^{\frac{5}{2}}}{5} - \frac{2 \cdot 0^{\frac{5}{2}}}{5})$

Step 9.2.3.12

Multiply $2$ by $2^{\frac{5}{2}}$ by adding the exponents.

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Step 9.2.3.12.1

Multiply $2$ by $2^{\frac{5}{2}}$ .

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Step 9.2.3.12.1.1

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

$\frac{2^{\frac{7}{2}}}{3} - (\frac{2^{1} \cdot 2^{\frac{5}{2}}}{5} - \frac{2 \cdot 0^{\frac{5}{2}}}{5})$

Step 9.2.3.12.1.2

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

$\frac{2^{\frac{7}{2}}}{3} - (\frac{2^{1 + \frac{5}{2}}}{5} - \frac{2 \cdot 0^{\frac{5}{2}}}{5})$

Step 9.2.3.12.2

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

$\frac{2^{\frac{7}{2}}}{3} - (\frac{2^{\frac{2}{2} + \frac{5}{2}}}{5} - \frac{2 \cdot 0^{\frac{5}{2}}}{5})$

Step 9.2.3.12.3

Combine the numerators over the common denominator.

$\frac{2^{\frac{7}{2}}}{3} - (\frac{2^{\frac{2 + 5}{2}}}{5} - \frac{2 \cdot 0^{\frac{5}{2}}}{5})$

Step 9.2.3.12.4

Add $2$ and $5$ .

$\frac{2^{\frac{7}{2}}}{3} - (\frac{2^{\frac{7}{2}}}{5} - \frac{2 \cdot 0^{\frac{5}{2}}}{5})$

Step 9.2.3.13

Rewrite $0$ as $0^{2}$ .

$\frac{2^{\frac{7}{2}}}{3} - (\frac{2^{\frac{7}{2}}}{5} - \frac{2 \cdot {(0^{2})}^{\frac{5}{2}}}{5})$

Step 9.2.3.14

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

$\frac{2^{\frac{7}{2}}}{3} - (\frac{2^{\frac{7}{2}}}{5} - \frac{2 \cdot 0^{2 (\frac{5}{2})}}{5})$

Step 9.2.3.15

Cancel the common factor of $2$ .

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Step 9.2.3.15.1

Cancel the common factor.

$\frac{2^{\frac{7}{2}}}{3} - (\frac{2^{\frac{7}{2}}}{5} - \frac{2 \cdot 0^{2 (\frac{5}{2})}}{5})$

Step 9.2.3.15.2

Rewrite the expression.

$\frac{2^{\frac{7}{2}}}{3} - (\frac{2^{\frac{7}{2}}}{5} - \frac{2 \cdot 0^{5}}{5})$

Step 9.2.3.16

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

$\frac{2^{\frac{7}{2}}}{3} - (\frac{2^{\frac{7}{2}}}{5} - \frac{2 \cdot 0}{5})$

Step 9.2.3.17

Multiply $2$ by $0$ .

$\frac{2^{\frac{7}{2}}}{3} - (\frac{2^{\frac{7}{2}}}{5} - \frac{0}{5})$

Step 9.2.3.18

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

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Step 9.2.3.18.1

Factor $5$ out of $0$ .

$\frac{2^{\frac{7}{2}}}{3} - (\frac{2^{\frac{7}{2}}}{5} - \frac{5 (0)}{5})$

Step 9.2.3.18.2

Cancel the common factors.

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Step 9.2.3.18.2.1

Factor $5$ out of $5$ .

$\frac{2^{\frac{7}{2}}}{3} - (\frac{2^{\frac{7}{2}}}{5} - \frac{5 \cdot 0}{5 \cdot 1})$

Step 9.2.3.18.2.2

Cancel the common factor.

$\frac{2^{\frac{7}{2}}}{3} - (\frac{2^{\frac{7}{2}}}{5} - \frac{5 \cdot 0}{5 \cdot 1})$

Step 9.2.3.18.2.3

Rewrite the expression.

$\frac{2^{\frac{7}{2}}}{3} - (\frac{2^{\frac{7}{2}}}{5} - \frac{0}{1})$

Step 9.2.3.18.2.4

Divide $0$ by $1$ .

$\frac{2^{\frac{7}{2}}}{3} - (\frac{2^{\frac{7}{2}}}{5} - 0)$

Step 9.2.3.19

Multiply $- 1$ by $0$ .

$\frac{2^{\frac{7}{2}}}{3} - (\frac{2^{\frac{7}{2}}}{5} + 0)$

Step 9.2.3.20

Add $\frac{2^{\frac{7}{2}}}{5}$ and $0$ .

$\frac{2^{\frac{7}{2}}}{3} - \frac{2^{\frac{7}{2}}}{5}$

Step 9.2.3.21

To write $\frac{2^{\frac{7}{2}}}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{2^{\frac{7}{2}}}{3} \cdot \frac{5}{5} - \frac{2^{\frac{7}{2}}}{5}$

Step 9.2.3.22

To write $- \frac{2^{\frac{7}{2}}}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2^{\frac{7}{2}}}{3} \cdot \frac{5}{5} - \frac{2^{\frac{7}{2}}}{5} \cdot \frac{3}{3}$

Step 9.2.3.23

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.2.3.23.1

Multiply $\frac{2^{\frac{7}{2}}}{3}$ by $\frac{5}{5}$ .

$\frac{2^{\frac{7}{2}} \cdot 5}{3 \cdot 5} - \frac{2^{\frac{7}{2}}}{5} \cdot \frac{3}{3}$

Step 9.2.3.23.2

Multiply $3$ by $5$ .

$\frac{2^{\frac{7}{2}} \cdot 5}{15} - \frac{2^{\frac{7}{2}}}{5} \cdot \frac{3}{3}$

Step 9.2.3.23.3

Multiply $\frac{2^{\frac{7}{2}}}{5}$ by $\frac{3}{3}$ .

$\frac{2^{\frac{7}{2}} \cdot 5}{15} - \frac{2^{\frac{7}{2}} \cdot 3}{5 \cdot 3}$

Step 9.2.3.23.4

Multiply $5$ by $3$ .

$\frac{2^{\frac{7}{2}} \cdot 5}{15} - \frac{2^{\frac{7}{2}} \cdot 3}{15}$

Step 9.2.3.24

Combine the numerators over the common denominator.

$\frac{2^{\frac{7}{2}} \cdot 5 - 2^{\frac{7}{2}} \cdot 3}{15}$

Step 9.2.3.25

Move $5$ to the left of $2^{\frac{7}{2}}$ .

$\frac{5 \cdot 2^{\frac{7}{2}} - 2^{\frac{7}{2}} \cdot 3}{15}$

Step 9.2.3.26

Multiply $3$ by $- 1$ .

$\frac{5 \cdot 2^{\frac{7}{2}} - 3 \cdot 2^{\frac{7}{2}}}{15}$

Step 9.2.3.27

Subtract $3 \cdot 2^{\frac{7}{2}}$ from $5 \cdot 2^{\frac{7}{2}}$ .

$\frac{2 \cdot 2^{\frac{7}{2}}}{15}$

Step 9.2.3.28

Multiply $2$ by $2^{\frac{7}{2}}$ by adding the exponents.

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Step 9.2.3.28.1

Multiply $2$ by $2^{\frac{7}{2}}$ .

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Step 9.2.3.28.1.1

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

$\frac{2^{1} \cdot 2^{\frac{7}{2}}}{15}$

Step 9.2.3.28.1.2

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

$\frac{2^{1 + \frac{7}{2}}}{15}$

Step 9.2.3.28.2

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

$\frac{2^{\frac{2}{2} + \frac{7}{2}}}{15}$

Step 9.2.3.28.3

Combine the numerators over the common denominator.

$\frac{2^{\frac{2 + 7}{2}}}{15}$

Step 9.2.3.28.4

Add $2$ and $7$ .

$\frac{2^{\frac{9}{2}}}{15}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{2^{\frac{9}{2}}}{15}$

Decimal Form:

$1.50849446 \dots$

Step 11