Calculus Examples

$\int_{0}^{3} (3 - t) \sqrt{t} d t$

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Factor out negative.

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Add $2$ and $1$ .

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

Since $3$ is constant with respect to $t$ , move $3$ out of the integral.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify the answer.

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Step 9.1

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

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

Step 9.2

Substitute and simplify.

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Step 9.2.1

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

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

Step 9.2.2

Evaluate $\frac{2 t^{\frac{5}{2}}}{5}$ at $3$ and at $0$ .

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

Step 9.2.3

Simplify.

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Step 9.2.3.1

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

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

Step 9.2.3.2

Multiply $3^{\frac{3}{2}}$ by $3^{- 1}$ by adding the exponents.

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Step 9.2.3.2.1

Move $3^{- 1}$ .

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

Step 9.2.3.2.2

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

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

Step 9.2.3.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 9.2.3.2.4

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 9.2.3.2.5

Combine the numerators over the common denominator.

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

Step 9.2.3.2.6

Simplify the numerator.

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Step 9.2.3.2.6.1

Multiply $- 1$ by $2$ .

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

Step 9.2.3.2.6.2

Add $- 2$ and $3$ .

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

Step 9.2.3.3

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

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

Step 9.2.3.4

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

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

Step 9.2.3.5

Cancel the common factor of $2$ .

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Step 9.2.3.5.1

Cancel the common factor.

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

Step 9.2.3.5.2

Rewrite the expression.

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

Step 9.2.3.6

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

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

Step 9.2.3.7

Multiply $2$ by $0$ .

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

Step 9.2.3.8

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

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Step 9.2.3.8.1

Factor $3$ out of $0$ .

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

Step 9.2.3.8.2

Cancel the common factors.

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Step 9.2.3.8.2.1

Factor $3$ out of $3$ .

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

Step 9.2.3.8.2.2

Cancel the common factor.

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

Step 9.2.3.8.2.3

Rewrite the expression.

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

Step 9.2.3.8.2.4

Divide $0$ by $1$ .

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

Step 9.2.3.9

Multiply $- 1$ by $0$ .

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

Step 9.2.3.10

Add $2 \cdot 3^{\frac{1}{2}}$ and $0$ .

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

Step 9.2.3.11

Multiply $2$ by $3$ .

$6 \cdot 3^{\frac{1}{2}} - (\frac{2 \cdot 3^{\frac{5}{2}}}{5} - \frac{2 \cdot 0^{\frac{5}{2}}}{5})$

Step 9.2.3.12

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

$6 \cdot 3^{\frac{1}{2}} - (\frac{2 \cdot 3^{\frac{5}{2}}}{5} - \frac{2 \cdot {(0^{2})}^{\frac{5}{2}}}{5})$

Step 9.2.3.13

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

$6 \cdot 3^{\frac{1}{2}} - (\frac{2 \cdot 3^{\frac{5}{2}}}{5} - \frac{2 \cdot 0^{2 (\frac{5}{2})}}{5})$

Step 9.2.3.14

Cancel the common factor of $2$ .

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Step 9.2.3.14.1

Cancel the common factor.

$6 \cdot 3^{\frac{1}{2}} - (\frac{2 \cdot 3^{\frac{5}{2}}}{5} - \frac{2 \cdot 0^{2 (\frac{5}{2})}}{5})$

Step 9.2.3.14.2

Rewrite the expression.

$6 \cdot 3^{\frac{1}{2}} - (\frac{2 \cdot 3^{\frac{5}{2}}}{5} - \frac{2 \cdot 0^{5}}{5})$

Step 9.2.3.15

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

$6 \cdot 3^{\frac{1}{2}} - (\frac{2 \cdot 3^{\frac{5}{2}}}{5} - \frac{2 \cdot 0}{5})$

Step 9.2.3.16

Multiply $2$ by $0$ .

$6 \cdot 3^{\frac{1}{2}} - (\frac{2 \cdot 3^{\frac{5}{2}}}{5} - \frac{0}{5})$

Step 9.2.3.17

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

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Step 9.2.3.17.1

Factor $5$ out of $0$ .

$6 \cdot 3^{\frac{1}{2}} - (\frac{2 \cdot 3^{\frac{5}{2}}}{5} - \frac{5 (0)}{5})$

Step 9.2.3.17.2

Cancel the common factors.

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Step 9.2.3.17.2.1

Factor $5$ out of $5$ .

$6 \cdot 3^{\frac{1}{2}} - (\frac{2 \cdot 3^{\frac{5}{2}}}{5} - \frac{5 \cdot 0}{5 \cdot 1})$

Step 9.2.3.17.2.2

Cancel the common factor.

$6 \cdot 3^{\frac{1}{2}} - (\frac{2 \cdot 3^{\frac{5}{2}}}{5} - \frac{5 \cdot 0}{5 \cdot 1})$

Step 9.2.3.17.2.3

Rewrite the expression.

$6 \cdot 3^{\frac{1}{2}} - (\frac{2 \cdot 3^{\frac{5}{2}}}{5} - \frac{0}{1})$

Step 9.2.3.17.2.4

Divide $0$ by $1$ .

$6 \cdot 3^{\frac{1}{2}} - (\frac{2 \cdot 3^{\frac{5}{2}}}{5} - 0)$

Step 9.2.3.18

Multiply $- 1$ by $0$ .

$6 \cdot 3^{\frac{1}{2}} - (\frac{2 \cdot 3^{\frac{5}{2}}}{5} + 0)$

Step 9.2.3.19

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

$6 \cdot 3^{\frac{1}{2}} - \frac{2 \cdot 3^{\frac{5}{2}}}{5}$

Step 9.2.3.20

To write $6 \cdot 3^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$6 \cdot 3^{\frac{1}{2}} \cdot \frac{5}{5} - \frac{2 \cdot 3^{\frac{5}{2}}}{5}$

Step 9.2.3.21

Combine $6 \cdot 3^{\frac{1}{2}}$ and $\frac{5}{5}$ .

$\frac{6 \cdot 3^{\frac{1}{2}} \cdot 5}{5} - \frac{2 \cdot 3^{\frac{5}{2}}}{5}$

Step 9.2.3.22

Combine the numerators over the common denominator.

$\frac{6 \cdot 3^{\frac{1}{2}} \cdot 5 - 2 \cdot 3^{\frac{5}{2}}}{5}$

Step 9.2.3.23

Multiply $5$ by $6$ .

$\frac{30 \cdot 3^{\frac{1}{2}} - 2 \cdot 3^{\frac{5}{2}}}{5}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{30 \cdot 3^{\frac{1}{2}} - 2 \cdot 3^{\frac{5}{2}}}{5}$

Decimal Form:

$4.15692193 \dots$

Step 11