Calculus Examples

$\int_{0}^{1} \frac{r^{3}}{\sqrt{4 + r^{2}}} d r$

Step 1

Let $u = 4 + r^{2}$ . Then $d u = 2 r d r$ , so $\frac{1}{2} d u = r d r$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 4 + r^{2}$ . Find $\frac{d u}{d r}$ .

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Step 1.1.1

Differentiate $4 + r^{2}$ .

$\frac{d}{d r} [4 + r^{2}]$

Step 1.1.2

By the Sum Rule, the derivative of $4 + r^{2}$ with respect to $r$ is $\frac{d}{d r} [4] + \frac{d}{d r} [r^{2}]$ .

$\frac{d}{d r} [4] + \frac{d}{d r} [r^{2}]$

Step 1.1.3

Since $4$ is constant with respect to $r$ , the derivative of $4$ with respect to $r$ is $0$ .

$0 + \frac{d}{d r} [r^{2}]$

Step 1.1.4

Differentiate using the Power Rule which states that $\frac{d}{d r} [r^{n}]$ is $n r^{n - 1}$ where $n = 2$ .

$0 + 2 r$

Step 1.1.5

Add $0$ and $2 r$ .

$2 r$

Step 1.2

Substitute the lower limit in for $r$ in $u = 4 + r^{2}$ .

$u_{lower} = 4 + 0^{2}$

Step 1.3

Simplify.

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Step 1.3.1

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

$u_{lower} = 4 + 0$

Step 1.3.2

Add $4$ and $0$ .

$u_{lower} = 4$

Step 1.4

Substitute the upper limit in for $r$ in $u = 4 + r^{2}$ .

$u_{upper} = 4 + 1^{2}$

Step 1.5

Simplify.

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Step 1.5.1

One to any power is one.

$u_{upper} = 4 + 1$

Step 1.5.2

Add $4$ and $1$ .

$u_{upper} = 5$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 4$

$u_{upper} = 5$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{4}^{5} \frac{{\sqrt{u - 4}}^{2}}{\sqrt{u}} \cdot \frac{1}{2} d u$

Step 2

Simplify.

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Step 2.1

Rewrite ${\sqrt{u - 4}}^{2}$ as $u - 4$ .

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Step 2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u - 4}$ as ${(u - 4)}^{\frac{1}{2}}$ .

$\int_{4}^{5} \frac{{({(u - 4)}^{\frac{1}{2}})}^{2}}{\sqrt{u}} \cdot \frac{1}{2} d u$

Step 2.1.2

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

$\int_{4}^{5} \frac{{(u - 4)}^{\frac{1}{2} \cdot 2}}{\sqrt{u}} \cdot \frac{1}{2} d u$

Step 2.1.3

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

$\int_{4}^{5} \frac{{(u - 4)}^{\frac{2}{2}}}{\sqrt{u}} \cdot \frac{1}{2} d u$

Step 2.1.4

Cancel the common factor of $2$ .

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Step 2.1.4.1

Cancel the common factor.

$\int_{4}^{5} \frac{{(u - 4)}^{\frac{2}{2}}}{\sqrt{u}} \cdot \frac{1}{2} d u$

Step 2.1.4.2

Rewrite the expression.

$\int_{4}^{5} \frac{{(u - 4)}^{1}}{\sqrt{u}} \cdot \frac{1}{2} d u$

Step 2.1.5

Simplify.

$\int_{4}^{5} \frac{u - 4}{\sqrt{u}} \cdot \frac{1}{2} d u$

Step 2.2

Multiply $\frac{u - 4}{\sqrt{u}}$ by $\frac{1}{2}$ .

$\int_{4}^{5} \frac{u - 4}{\sqrt{u} \cdot 2} d u$

Step 2.3

Move $2$ to the left of $\sqrt{u}$ .

$\int_{4}^{5} \frac{u - 4}{2 \sqrt{u}} d u$

Step 3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{4}^{5} \frac{u - 4}{\sqrt{u}} d u$

Step 4

Apply basic rules of exponents.

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Step 4.1

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

$\frac{1}{2} \int_{4}^{5} \frac{u - 4}{u^{\frac{1}{2}}} d u$

Step 4.2

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

$\frac{1}{2} \int_{4}^{5} (u - 4) {(u^{\frac{1}{2}})}^{- 1} d u$

Step 4.3

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

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Step 4.3.1

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

$\frac{1}{2} \int_{4}^{5} (u - 4) u^{\frac{1}{2} \cdot - 1} d u$

Step 4.3.2

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

$\frac{1}{2} \int_{4}^{5} (u - 4) u^{\frac{- 1}{2}} d u$

Step 4.3.3

Move the negative in front of the fraction.

$\frac{1}{2} \int_{4}^{5} (u - 4) u^{- \frac{1}{2}} d u$

Step 5

Expand $(u - 4) u^{- \frac{1}{2}}$ .

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Step 5.1

Apply the distributive property.

$\frac{1}{2} \int_{4}^{5} u \cdot u^{- \frac{1}{2}} - 4 u^{- \frac{1}{2}} d u$

Step 5.2

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

$\frac{1}{2} \int_{4}^{5} u^{1} u^{- \frac{1}{2}} - 4 u^{- \frac{1}{2}} d u$

Step 5.3

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

$\frac{1}{2} \int_{4}^{5} u^{1 - \frac{1}{2}} - 4 u^{- \frac{1}{2}} d u$

Step 5.4

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

$\frac{1}{2} \int_{4}^{5} u^{\frac{2}{2} - \frac{1}{2}} - 4 u^{- \frac{1}{2}} d u$

Step 5.5

Combine the numerators over the common denominator.

$\frac{1}{2} \int_{4}^{5} u^{\frac{2 - 1}{2}} - 4 u^{- \frac{1}{2}} d u$

Step 5.6

Subtract $1$ from $2$ .

$\frac{1}{2} \int_{4}^{5} u^{\frac{1}{2}} - 4 u^{- \frac{1}{2}} d u$

Step 6

Split the single integral into multiple integrals.

$\frac{1}{2} (\int_{4}^{5} u^{\frac{1}{2}} d u + \int_{4}^{5} - 4 u^{- \frac{1}{2}} d u)$

Step 7

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

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

Step 8

Since $- 4$ is constant with respect to $u$ , move $- 4$ out of the integral.

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

Step 9

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

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

Step 10

Combine fractions.

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Step 10.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $5$ and at $4$ .

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

Simplify the expression.

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Step 10.4.1

Simplify.

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Step 10.4.1.1

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

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

Step 10.4.1.2

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

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

Step 10.4.1.3

Cancel the common factor of $2$ .

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Step 10.4.1.3.1

Cancel the common factor.

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

Step 10.4.1.3.2

Rewrite the expression.

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

Step 10.4.1.4

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

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

Step 10.4.2

Simplify.

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Step 10.4.2.1

Multiply $8$ by $- 1$ .

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

Step 10.4.2.2

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

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

Step 10.4.2.3

Multiply $- 8$ by $2$ .

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

Step 10.4.2.4

Move the negative in front of the fraction.

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

Step 10.4.3

Simplify.

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Step 10.4.3.1

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

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

Step 10.4.3.2

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

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

Step 10.4.3.3

Cancel the common factor of $2$ .

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Step 10.4.3.3.1

Cancel the common factor.

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

Step 10.4.3.3.2

Rewrite the expression.

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

Step 10.4.3.4

Evaluate the exponent.

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

Step 10.4.4

Multiply $- 2$ by $2$ .

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

Step 10.4.5

Simplify.

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Step 10.4.5.1

To write $- 4 (2 \cdot 5^{\frac{1}{2}} - 4)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 10.4.5.2

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

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

Step 10.4.5.3

Combine the numerators over the common denominator.

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

Step 10.4.5.4

Multiply $3$ by $- 4$ .

$\frac{1}{2} (\frac{2 \cdot 5^{\frac{3}{2}} - 12 (2 \cdot 5^{\frac{1}{2}} - 4)}{3} - \frac{16}{3})$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2} \cdot (\frac{2 \cdot 5^{\frac{3}{2}} - 12 (2 \cdot 5^{\frac{1}{2}} - 4)}{3} - \frac{16}{3})$

Decimal Form:

$0.11584138 \dots$