Calculus Examples

$\int_{1}^{4} (1 - u) \sqrt{u} d u$

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

$\int_{1}^{4} 1 u^{\frac{1}{2}} - u \cdot u^{\frac{1}{2}} d u$

Step 2.2

Multiply $u^{\frac{1}{2}}$ by $1$ .

$\int_{1}^{4} u^{\frac{1}{2}} - u \cdot u^{\frac{1}{2}} d u$

Step 2.3

Factor out negative.

$\int_{1}^{4} u^{\frac{1}{2}} - (u \cdot u^{\frac{1}{2}}) d u$

Step 2.4

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

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

Step 2.5

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

$\int_{1}^{4} u^{\frac{1}{2}} - u^{1 + \frac{1}{2}} d u$

Step 2.6

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

$\int_{1}^{4} u^{\frac{1}{2}} - u^{\frac{2}{2} + \frac{1}{2}} d u$

Step 2.7

Combine the numerators over the common denominator.

$\int_{1}^{4} u^{\frac{1}{2}} - u^{\frac{2 + 1}{2}} d u$

Step 2.8

Add $2$ and $1$ .

$\int_{1}^{4} u^{\frac{1}{2}} - u^{\frac{3}{2}} d u$

Step 3

Split the single integral into multiple integrals.

$\int_{1}^{4} u^{\frac{1}{2}} d u + \int_{1}^{4} - u^{\frac{3}{2}} d u$

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify the answer.

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Step 7.1

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

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

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $4$ and at $1$ .

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

Step 7.2.2

Evaluate $\frac{2 u^{\frac{5}{2}}}{5}$ at $4$ and at $1$ .

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

Step 7.2.3

Simplify.

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Step 7.2.3.1

Rewrite $4$ as $2^{2}$ .

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

Step 7.2.3.2

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

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

Step 7.2.3.3

Cancel the common factor of $2$ .

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Step 7.2.3.3.1

Cancel the common factor.

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

Step 7.2.3.3.2

Rewrite the expression.

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

Step 7.2.3.4

Raise $2$ to the power of $3$ .

$\frac{2}{3} \cdot 8 - \frac{2}{3} \cdot 1^{\frac{3}{2}} - (\frac{2 \cdot 4^{\frac{5}{2}}}{5} - \frac{2 \cdot 1^{\frac{5}{2}}}{5})$

Step 7.2.3.5

Combine $\frac{2}{3}$ and $8$ .

$\frac{2 \cdot 8}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}} - (\frac{2 \cdot 4^{\frac{5}{2}}}{5} - \frac{2 \cdot 1^{\frac{5}{2}}}{5})$

Step 7.2.3.6

Multiply $2$ by $8$ .

$\frac{16}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}} - (\frac{2 \cdot 4^{\frac{5}{2}}}{5} - \frac{2 \cdot 1^{\frac{5}{2}}}{5})$

Step 7.2.3.7

One to any power is one.

$\frac{16}{3} - \frac{2}{3} \cdot 1 - (\frac{2 \cdot 4^{\frac{5}{2}}}{5} - \frac{2 \cdot 1^{\frac{5}{2}}}{5})$

Step 7.2.3.8

Multiply $- 1$ by $1$ .

$\frac{16}{3} - \frac{2}{3} - (\frac{2 \cdot 4^{\frac{5}{2}}}{5} - \frac{2 \cdot 1^{\frac{5}{2}}}{5})$

Step 7.2.3.9

Combine the numerators over the common denominator.

$\frac{16 - 2}{3} - (\frac{2 \cdot 4^{\frac{5}{2}}}{5} - \frac{2 \cdot 1^{\frac{5}{2}}}{5})$

Step 7.2.3.10

Subtract $2$ from $16$ .

$\frac{14}{3} - (\frac{2 \cdot 4^{\frac{5}{2}}}{5} - \frac{2 \cdot 1^{\frac{5}{2}}}{5})$

Step 7.2.3.11

Rewrite $4$ as $2^{2}$ .

$\frac{14}{3} - (\frac{2 \cdot {(2^{2})}^{\frac{5}{2}}}{5} - \frac{2 \cdot 1^{\frac{5}{2}}}{5})$

Step 7.2.3.12

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

$\frac{14}{3} - (\frac{2 \cdot 2^{2 (\frac{5}{2})}}{5} - \frac{2 \cdot 1^{\frac{5}{2}}}{5})$

Step 7.2.3.13

Cancel the common factor of $2$ .

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Step 7.2.3.13.1

Cancel the common factor.

$\frac{14}{3} - (\frac{2 \cdot 2^{2 (\frac{5}{2})}}{5} - \frac{2 \cdot 1^{\frac{5}{2}}}{5})$

Step 7.2.3.13.2

Rewrite the expression.

$\frac{14}{3} - (\frac{2 \cdot 2^{5}}{5} - \frac{2 \cdot 1^{\frac{5}{2}}}{5})$

Step 7.2.3.14

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

$\frac{14}{3} - (\frac{2 \cdot 32}{5} - \frac{2 \cdot 1^{\frac{5}{2}}}{5})$

Step 7.2.3.15

Multiply $2$ by $32$ .

$\frac{14}{3} - (\frac{64}{5} - \frac{2 \cdot 1^{\frac{5}{2}}}{5})$

Step 7.2.3.16

One to any power is one.

$\frac{14}{3} - (\frac{64}{5} - \frac{2 \cdot 1}{5})$

Step 7.2.3.17

Multiply $2$ by $1$ .

$\frac{14}{3} - (\frac{64}{5} - \frac{2}{5})$

Step 7.2.3.18

Combine the numerators over the common denominator.

$\frac{14}{3} - \frac{64 - 2}{5}$

Step 7.2.3.19

Subtract $2$ from $64$ .

$\frac{14}{3} - \frac{62}{5}$

Step 7.2.3.20

To write $\frac{14}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{14}{3} \cdot \frac{5}{5} - \frac{62}{5}$

Step 7.2.3.21

To write $- \frac{62}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{14}{3} \cdot \frac{5}{5} - \frac{62}{5} \cdot \frac{3}{3}$

Step 7.2.3.22

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

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Step 7.2.3.22.1

Multiply $\frac{14}{3}$ by $\frac{5}{5}$ .

$\frac{14 \cdot 5}{3 \cdot 5} - \frac{62}{5} \cdot \frac{3}{3}$

Step 7.2.3.22.2

Multiply $3$ by $5$ .

$\frac{14 \cdot 5}{15} - \frac{62}{5} \cdot \frac{3}{3}$

Step 7.2.3.22.3

Multiply $\frac{62}{5}$ by $\frac{3}{3}$ .

$\frac{14 \cdot 5}{15} - \frac{62 \cdot 3}{5 \cdot 3}$

Step 7.2.3.22.4

Multiply $5$ by $3$ .

$\frac{14 \cdot 5}{15} - \frac{62 \cdot 3}{15}$

Step 7.2.3.23

Combine the numerators over the common denominator.

$\frac{14 \cdot 5 - 62 \cdot 3}{15}$

Step 7.2.3.24

Simplify the numerator.

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Step 7.2.3.24.1

Multiply $14$ by $5$ .

$\frac{70 - 62 \cdot 3}{15}$

Step 7.2.3.24.2

Multiply $- 62$ by $3$ .

$\frac{70 - 186}{15}$

Step 7.2.3.24.3

Subtract $186$ from $70$ .

$\frac{- 116}{15}$

Step 7.2.3.25

Move the negative in front of the fraction.

$- \frac{116}{15}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{116}{15}$

Decimal Form:

$- 7.7\overline{3}$

Mixed Number Form:

$- 7 \frac{11}{15}$

Step 9