Calculus Examples

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

Step 1

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

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

Step 2

Move $u^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 3

Multiply the exponents in ${(u^{\frac{1}{2}})}^{- 1}$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Move the negative in front of the fraction.

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

Step 4

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

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Step 4.1

Apply the distributive property.

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Subtract $1$ from $2$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Substitute and simplify.

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

Simplify.

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Step 9.3.1

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

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

Step 9.3.2

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

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

Step 9.3.3

Cancel the common factor of $2$ .

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Step 9.3.3.1

Cancel the common factor.

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

Step 9.3.3.2

Rewrite the expression.

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

Step 9.3.4

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

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

Step 9.3.5

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

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

Step 9.3.6

Multiply $2$ by $8$ .

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

Step 9.3.7

One to any power is one.

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

Step 9.3.8

Multiply $- 1$ by $1$ .

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

Step 9.3.9

Combine the numerators over the common denominator.

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

Step 9.3.10

Subtract $2$ from $16$ .

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

Step 9.3.11

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

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

Step 9.3.12

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

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

Step 9.3.13

Cancel the common factor of $2$ .

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Step 9.3.13.1

Cancel the common factor.

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

Step 9.3.13.2

Rewrite the expression.

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

Step 9.3.14

Evaluate the exponent.

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

Step 9.3.15

Multiply $2$ by $2$ .

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

Step 9.3.16

One to any power is one.

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

Step 9.3.17

Multiply $- 2$ by $1$ .

$\frac{14}{3} - 2 (4 - 2)$

Step 9.3.18

Subtract $2$ from $4$ .

$\frac{14}{3} - 2 \cdot 2$

Step 9.3.19

Multiply $- 2$ by $2$ .

$\frac{14}{3} - 4$

Step 9.3.20

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

$\frac{14}{3} - 4 \cdot \frac{3}{3}$

Step 9.3.21

Combine $- 4$ and $\frac{3}{3}$ .

$\frac{14}{3} + \frac{- 4 \cdot 3}{3}$

Step 9.3.22

Combine the numerators over the common denominator.

$\frac{14 - 4 \cdot 3}{3}$

Step 9.3.23

Simplify the numerator.

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Step 9.3.23.1

Multiply $- 4$ by $3$ .

$\frac{14 - 12}{3}$

Step 9.3.23.2

Subtract $12$ from $14$ .

$\frac{2}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$

Step 11