Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

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

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

Step 1.1.2

Factor $y^{\frac{1}{2}}$ out of $y^{\frac{1}{2}} - y$ .

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Step 1.1.2.1

Multiply by $1$ .

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

Step 1.1.2.2

Factor $y^{\frac{1}{2}}$ out of $- y$ .

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

Step 1.1.2.3

Factor $y^{\frac{1}{2}}$ out of $y^{\frac{1}{2}} \cdot 1 + y^{\frac{1}{2}} (- y^{\frac{1}{2}})$ .

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

Step 1.2

Move $y^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

Combine the numerators over the common denominator.

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

Step 1.3.5

Simplify the numerator.

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Step 1.3.5.1

Multiply $2$ by $2$ .

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

Step 1.3.5.2

Subtract $1$ from $4$ .

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

Step 2

Move $y^{\frac{3}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\int_{1}^{4} (1 - y^{\frac{1}{2}}) {(y^{\frac{3}{2}})}^{- 1} d y$

Step 3

Multiply the exponents in ${(y^{\frac{3}{2}})}^{- 1}$ .

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Step 3.1

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

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

Step 3.2

Multiply $\frac{3}{2} \cdot - 1$ .

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Step 3.2.1

Combine $\frac{3}{2}$ and $- 1$ .

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

Step 3.2.2

Multiply $3$ by $- 1$ .

$\int_{1}^{4} (1 - y^{\frac{1}{2}}) y^{\frac{- 3}{2}} d y$

Step 3.3

Move the negative in front of the fraction.

$\int_{1}^{4} (1 - y^{\frac{1}{2}}) y^{- \frac{3}{2}} d y$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $y^{- \frac{3}{2}}$ by $1$ .

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

Step 4.3

Factor out negative.

$\int_{1}^{4} y^{- \frac{3}{2}} - (y^{\frac{1}{2}} y^{- \frac{3}{2}}) d y$

Step 4.4

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

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Subtract $3$ from $1$ .

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

Step 4.7

Cancel the common factor of $- 2$ and $2$ .

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Step 4.7.1

Factor $2$ out of $- 2$ .

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

Step 4.7.2

Cancel the common factors.

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Step 4.7.2.1

Factor $2$ out of $2$ .

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

Step 4.7.2.2

Cancel the common factor.

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

Step 4.7.2.3

Rewrite the expression.

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

Step 4.7.2.4

Divide $- 1$ by $1$ .

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

Step 4.8

Reorder $y^{- \frac{3}{2}}$ and $- y^{- 1}$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

The integral of $y^{- 1}$ with respect to $y$ is $\ln (| y |)$ .

$- (\ln (| y |)]_{1}^{4}) + \int_{1}^{4} y^{- \frac{3}{2}} d y$

Step 8

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

$- (\ln (| y |)]_{1}^{4}) + - 2 y^{- \frac{1}{2}}]_{1}^{4}$

Step 9

Simplify the answer.

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Step 9.1

Substitute and simplify.

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Step 9.1.1

Evaluate $\ln (| y |)$ at $4$ and at $1$ .

$- ((\ln (| 4 |)) - \ln (| 1 |)) + - 2 y^{- \frac{1}{2}}]_{1}^{4}$

Step 9.1.2

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

$- (\ln (| 4 |) - \ln (| 1 |)) + - 2 \cdot 4^{- \frac{1}{2}} + 2 \cdot 1^{- \frac{1}{2}}$

Step 9.1.3

Simplify.

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Step 9.1.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- (\ln (| 4 |) - \ln (| 1 |)) - 2 \cdot \frac{1}{4^{\frac{1}{2}}} + 2 \cdot 1^{- \frac{1}{2}}$

Step 9.1.3.2

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

$- (\ln (| 4 |) - \ln (| 1 |)) - 2 \cdot \frac{1}{{(2^{2})}^{\frac{1}{2}}} + 2 \cdot 1^{- \frac{1}{2}}$

Step 9.1.3.3

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

$- (\ln (| 4 |) - \ln (| 1 |)) - 2 \cdot \frac{1}{2^{2 (\frac{1}{2})}} + 2 \cdot 1^{- \frac{1}{2}}$

Step 9.1.3.4

Cancel the common factor of $2$ .

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Step 9.1.3.4.1

Cancel the common factor.

$- (\ln (| 4 |) - \ln (| 1 |)) - 2 \cdot \frac{1}{2^{2 (\frac{1}{2})}} + 2 \cdot 1^{- \frac{1}{2}}$

Step 9.1.3.4.2

Rewrite the expression.

$- (\ln (| 4 |) - \ln (| 1 |)) - 2 \cdot \frac{1}{2^{1}} + 2 \cdot 1^{- \frac{1}{2}}$

Step 9.1.3.5

Evaluate the exponent.

$- (\ln (| 4 |) - \ln (| 1 |)) - 2 \cdot \frac{1}{2} + 2 \cdot 1^{- \frac{1}{2}}$

Step 9.1.3.6

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

$- (\ln (| 4 |) - \ln (| 1 |)) + \frac{- 2}{2} + 2 \cdot 1^{- \frac{1}{2}}$

Step 9.1.3.7

Cancel the common factor of $- 2$ and $2$ .

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Step 9.1.3.7.1

Factor $2$ out of $- 2$ .

$- (\ln (| 4 |) - \ln (| 1 |)) + \frac{2 \cdot - 1}{2} + 2 \cdot 1^{- \frac{1}{2}}$

Step 9.1.3.7.2

Cancel the common factors.

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Step 9.1.3.7.2.1

Factor $2$ out of $2$ .

$- (\ln (| 4 |) - \ln (| 1 |)) + \frac{2 \cdot - 1}{2 (1)} + 2 \cdot 1^{- \frac{1}{2}}$

Step 9.1.3.7.2.2

Cancel the common factor.

$- (\ln (| 4 |) - \ln (| 1 |)) + \frac{2 \cdot - 1}{2 \cdot 1} + 2 \cdot 1^{- \frac{1}{2}}$

Step 9.1.3.7.2.3

Rewrite the expression.

$- (\ln (| 4 |) - \ln (| 1 |)) + \frac{- 1}{1} + 2 \cdot 1^{- \frac{1}{2}}$

Step 9.1.3.7.2.4

Divide $- 1$ by $1$ .

$- (\ln (| 4 |) - \ln (| 1 |)) - 1 + 2 \cdot 1^{- \frac{1}{2}}$

Step 9.1.3.8

One to any power is one.

$- (\ln (| 4 |) - \ln (| 1 |)) - 1 + 2 \cdot 1$

Step 9.1.3.9

Multiply $2$ by $1$ .

$- (\ln (| 4 |) - \ln (| 1 |)) - 1 + 2$

Step 9.1.3.10

Add $- 1$ and $2$ .

$- (\ln (| 4 |) - \ln (| 1 |)) + 1$

Step 9.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$- \ln (\frac{| 4 |}{| 1 |}) + 1$

Step 9.3

Simplify.

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Step 9.3.1

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$- \ln (\frac{4}{| 1 |}) + 1$

Step 9.3.2

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$- \ln (\frac{4}{1}) + 1$

Step 9.3.3

Divide $4$ by $1$ .

$- \ln (4) + 1$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \ln (4) + 1$

Decimal Form:

$- 0.38629436 \dots$

Step 11