Calculus Examples

$\int_{0}^{a} \frac{x^{3}}{\sqrt{a^{4} + x^{4}}} d x$

Step 1

Let $u = a^{4} + x^{4}$ . Then $d u = 4 x^{3} d x$ , so $\frac{1}{4} d u = x^{3} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = a^{4} + x^{4}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $a^{4} + x^{4}$ .

$\frac{d}{d x} [a^{4} + x^{4}]$

Step 1.1.2

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

$\frac{d}{d x} [a^{4}] + \frac{d}{d x} [x^{4}]$

Step 1.1.3

Since $a^{4}$ is constant with respect to $x$ , the derivative of $a^{4}$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x^{4}]$

Step 1.1.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$0 + 4 x^{3}$

Step 1.1.5

Add $0$ and $4 x^{3}$ .

$4 x^{3}$

Step 1.2

Substitute the lower limit in for $x$ in $u = a^{4} + x^{4}$ .

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

Step 1.3

Simplify.

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Step 1.3.1

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

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

Step 1.3.2

Add $a^{4}$ and $0$ .

$u_{lower} = a^{4}$

Step 1.4

Substitute the upper limit in for $x$ in $u = a^{4} + x^{4}$ .

$u_{upper} = a^{4} + a^{4}$

Step 1.5

Add $a^{4}$ and $a^{4}$ .

$u_{upper} = 2 a^{4}$

Step 1.6

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

$u_{lower} = a^{4}$

$u_{upper} = 2 a^{4}$

Step 1.7

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

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

Move $4$ to the left of $\sqrt{u}$ .

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

Step 3

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

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

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

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

Step 4.2

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.3.2

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

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

Step 4.3.3

Move the negative in front of the fraction.

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

Step 5

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

$\frac{1}{4} 2 u^{\frac{1}{2}}]_{a^{4}}^{2 a^{4}}$

Step 6

Substitute and simplify.

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Step 6.1

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

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

Step 6.2

Simplify.

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Step 6.2.1

Factor $2$ out of $2 a^{4}$ .

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

Step 6.2.2

Apply the product rule to $2 (a^{4})$ .

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

Step 6.2.3

Multiply the exponents in ${(a^{4})}^{\frac{1}{2}}$ .

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Step 6.2.3.1

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

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

Step 6.2.3.2

Cancel the common factor of $2$ .

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Step 6.2.3.2.1

Factor $2$ out of $4$ .

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

Step 6.2.3.2.2

Cancel the common factor.

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

Step 6.2.3.2.3

Rewrite the expression.

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

Step 6.2.4

Raise $2$ to the power of $1$ .

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

Step 6.2.5

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

$\frac{1}{4} (2^{\frac{1}{2} + 1} a^{2} - 2 {(a^{4})}^{\frac{1}{2}})$

Step 6.2.6

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

$\frac{1}{4} (2^{\frac{1}{2} + \frac{2}{2}} a^{2} - 2 {(a^{4})}^{\frac{1}{2}})$

Step 6.2.7

Combine the numerators over the common denominator.

$\frac{1}{4} (2^{\frac{1 + 2}{2}} a^{2} - 2 {(a^{4})}^{\frac{1}{2}})$

Step 6.2.8

Add $1$ and $2$ .

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

Step 6.2.9

Multiply the exponents in ${(a^{4})}^{\frac{1}{2}}$ .

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Step 6.2.9.1

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

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

Step 6.2.9.2

Cancel the common factor of $2$ .

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Step 6.2.9.2.1

Factor $2$ out of $4$ .

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

Step 6.2.9.2.2

Cancel the common factor.

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

Step 6.2.9.2.3

Rewrite the expression.

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

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Multiply $\frac{1}{4} (2^{\frac{3}{2}} a^{2})$ .

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Step 7.2.1

Combine $2^{\frac{3}{2}}$ and $\frac{1}{4}$ .

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

Step 7.2.2

Combine $\frac{2^{\frac{3}{2}}}{4}$ and $a^{2}$ .

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

Step 7.3

Cancel the common factor of $2$ .

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Step 7.3.1

Factor $2$ out of $4$ .

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

Step 7.3.2

Factor $2$ out of $- 2 a^{2}$ .

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

Step 7.3.3

Cancel the common factor.

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

Step 7.3.4

Rewrite the expression.

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

Step 7.4

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

$\frac{2^{\frac{3}{2}} a^{2}}{4} - \frac{a^{2}}{2}$