Calculus Examples

$\frac{x^{3}}{\sqrt{2 - x^{2}}}$

Step 1

Write $\frac{x^{3}}{\sqrt{2 - x^{2}}}$ as a function.

$f (x) = \frac{x^{3}}{\sqrt{2 - x^{2}}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \frac{x^{3}}{\sqrt{2 - x^{2}}} d x$

Step 4

Let $x = \sqrt{2} \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \sqrt{2} \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\sqrt{2} \cos (t)$ is positive.

$\int \frac{{(\sqrt{2} \sin (t))}^{3}}{\sqrt{{\sqrt{2}}^{2} - {(\sqrt{2} \sin (t))}^{2}}} (\sqrt{2} \cos (t)) d t$

Step 5

Simplify terms.

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Step 5.1

Simplify $\sqrt{{\sqrt{2}}^{2} - {(\sqrt{2} \sin (t))}^{2}}$ .

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Step 5.1.1

Apply the product rule to $\sqrt{2} \sin (t)$ .

$\int \frac{{(\sqrt{2} \sin (t))}^{3}}{\sqrt{{\sqrt{2}}^{2} - ({\sqrt{2}}^{2} \sin^{2} (t))}} (\sqrt{2} \cos (t)) d t$

Step 5.1.2

Multiply by $1$ .

$\int \frac{{(\sqrt{2} \sin (t))}^{3}}{\sqrt{{\sqrt{2}}^{2} \cdot 1 - ({\sqrt{2}}^{2} \sin^{2} (t))}} (\sqrt{2} \cos (t)) d t$

Step 5.1.3

Factor ${\sqrt{2}}^{2}$ out of $- ({\sqrt{2}}^{2} \sin^{2} (t))$ .

$\int \frac{{(\sqrt{2} \sin (t))}^{3}}{\sqrt{{\sqrt{2}}^{2} \cdot 1 + {\sqrt{2}}^{2} (- (\sin^{2} (t)))}} (\sqrt{2} \cos (t)) d t$

Step 5.1.4

Factor ${\sqrt{2}}^{2}$ out of ${\sqrt{2}}^{2} \cdot 1 + {\sqrt{2}}^{2} (- (\sin^{2} (t)))$ .

$\int \frac{{(\sqrt{2} \sin (t))}^{3}}{\sqrt{{\sqrt{2}}^{2} \cdot (1 - (\sin^{2} (t)))}} (\sqrt{2} \cos (t)) d t$

Step 5.1.5

Apply pythagorean identity.

$\int \frac{{(\sqrt{2} \sin (t))}^{3}}{\sqrt{{\sqrt{2}}^{2} \cdot \cos^{2} (t)}} (\sqrt{2} \cos (t)) d t$

Step 5.1.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 5.1.6.1

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

$\int \frac{{(\sqrt{2} \sin (t))}^{3}}{\sqrt{{(2^{\frac{1}{2}})}^{2} \cdot \cos^{2} (t)}} (\sqrt{2} \cos (t)) d t$

Step 5.1.6.2

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

$\int \frac{{(\sqrt{2} \sin (t))}^{3}}{\sqrt{2^{\frac{1}{2} \cdot 2} \cdot \cos^{2} (t)}} (\sqrt{2} \cos (t)) d t$

Step 5.1.6.3

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

$\int \frac{{(\sqrt{2} \sin (t))}^{3}}{\sqrt{2^{\frac{2}{2}} \cdot \cos^{2} (t)}} (\sqrt{2} \cos (t)) d t$

Step 5.1.6.4

Cancel the common factor of $2$ .

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Step 5.1.6.4.1

Cancel the common factor.

$\int \frac{{(\sqrt{2} \sin (t))}^{3}}{\sqrt{2^{\frac{2}{2}} \cdot \cos^{2} (t)}} (\sqrt{2} \cos (t)) d t$

Step 5.1.6.4.2

Rewrite the expression.

$\int \frac{{(\sqrt{2} \sin (t))}^{3}}{\sqrt{2^{1} \cdot \cos^{2} (t)}} (\sqrt{2} \cos (t)) d t$

Step 5.1.6.5

Evaluate the exponent.

$\int \frac{{(\sqrt{2} \sin (t))}^{3}}{\sqrt{2 \cdot \cos^{2} (t)}} (\sqrt{2} \cos (t)) d t$

$\int \frac{{(\sqrt{2} \sin (t))}^{3}}{\sqrt{2 \cos^{2} (t)}} (\sqrt{2} \cos (t)) d t$

Step 5.1.7

Reorder $2$ and $\cos^{2} (t)$ .

$\int \frac{{(\sqrt{2} \sin (t))}^{3}}{\sqrt{\cos^{2} (t) \cdot 2}} (\sqrt{2} \cos (t)) d t$

Step 5.1.8

Pull terms out from under the radical.

$\int \frac{{(\sqrt{2} \sin (t))}^{3}}{\cos (t) \sqrt{2}} (\sqrt{2} \cos (t)) d t$

Step 5.2

Reduce the expression by cancelling the common factors.

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Step 5.2.1

Cancel the common factor of $\sqrt{2} \cos (t)$ .

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Step 5.2.1.1

Factor $\sqrt{2} \cos (t)$ out of $\cos (t) \sqrt{2}$ .

$\int \frac{{(\sqrt{2} \sin (t))}^{3}}{\sqrt{2} \cos (t) (1)} (\sqrt{2} \cos (t)) d t$

Step 5.2.1.2

Cancel the common factor.

$\int \frac{{(\sqrt{2} \sin (t))}^{3}}{\sqrt{2} \cos (t) \cdot 1} (\sqrt{2} \cos (t)) d t$

Step 5.2.1.3

Rewrite the expression.

$\int {(\sqrt{2} \sin (t))}^{3} d t$

Step 5.2.2

Simplify the expression.

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Step 5.2.2.1

Apply the product rule to $\sqrt{2} \sin (t)$ .

$\int {\sqrt{2}}^{3} \sin^{3} (t) d t$

Step 5.2.2.2

Simplify.

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Step 5.2.2.2.1

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

$\int \sqrt{2^{3}} \sin^{3} (t) d t$

Step 5.2.2.2.2

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

$\int \sqrt{8} \sin^{3} (t) d t$

Step 6

Since $\sqrt{8}$ is constant with respect to $t$ , move $\sqrt{8}$ out of the integral.

$\sqrt{8} \int \sin^{3} (t) d t$

Step 7

Factor out $\sin^{2} (t)$ .

$\sqrt{8} \int \sin^{2} (t) \sin (t) d t$

Step 8

Using the Pythagorean Identity, rewrite $\sin^{2} (t)$ as $1 - \cos^{2} (t)$ .

$\sqrt{8} \int (1 - \cos^{2} (t)) \sin (t) d t$

Step 9

Let $u = \cos (t)$ . Then $d u = - \sin (t) d t$ , so $- \frac{1}{\sin (t)} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 9.1

Let $u = \cos (t)$ . Find $\frac{d u}{d t}$ .

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Step 9.1.1

Differentiate $\cos (t)$ .

$\frac{d}{d t} [\cos (t)]$

Step 9.1.2

The derivative of $\cos (t)$ with respect to $t$ is $- \sin (t)$ .

$- \sin (t)$

Step 9.2

Rewrite the problem using $u$ and $d u$ .

$\sqrt{8} \int - 1 + u^{2} d u$

Step 10

Split the single integral into multiple integrals.

$\sqrt{8} (\int - 1 d u + \int u^{2} d u)$

Step 11

Apply the constant rule.

$\sqrt{8} (- u + C + \int u^{2} d u)$

Step 12

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

$\sqrt{8} (- u + C + \frac{1}{3} u^{3} + C)$

Step 13

Simplify.

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Step 13.1

Combine $\frac{1}{3}$ and $u^{3}$ .

$\sqrt{8} (- u + C + \frac{u^{3}}{3} + C)$

Step 13.2

Simplify.

$2 \sqrt{2} (- u + \frac{u^{3}}{3}) + C$

Step 14

Substitute back in for each integration substitution variable.

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Step 14.1

Replace all occurrences of $u$ with $\cos (t)$ .

$2 \sqrt{2} (- \cos (t) + \frac{\cos^{3} (t)}{3}) + C$

Step 14.2

Replace all occurrences of $t$ with $\arcsin (\frac{x}{\sqrt{2}})$ .

$2 \sqrt{2} (- \cos (\arcsin (\frac{x}{\sqrt{2}})) + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15

Simplify.

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Step 15.1

Simplify each term.

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Step 15.1.1

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{x}{\sqrt{2}})}^{2}}, \frac{x}{\sqrt{2}})$ , $(\sqrt{1^{2} - {(\frac{x}{\sqrt{2}})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{x}{\sqrt{2}})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{x}{\sqrt{2}})}^{2}}, \frac{x}{\sqrt{2}})$ . Therefore, $\cos (\arcsin (\frac{x}{\sqrt{2}}))$ is $\sqrt{1 - {(\frac{x}{\sqrt{2}})}^{2}}$ .

$2 \sqrt{2} (- \sqrt{1 - {(\frac{x}{\sqrt{2}})}^{2}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.2

Rewrite $1$ as $1^{2}$ .

$2 \sqrt{2} (- \sqrt{1^{2} - {(\frac{x}{\sqrt{2}})}^{2}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \frac{x}{\sqrt{2}}$ .

$2 \sqrt{2} (- \sqrt{(1 + \frac{x}{\sqrt{2}}) (1 - \frac{x}{\sqrt{2}})} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.4

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

$2 \sqrt{2} (- \sqrt{(\frac{\sqrt{2}}{\sqrt{2}} + \frac{x}{\sqrt{2}}) (1 - \frac{x}{\sqrt{2}})} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.5

Combine the numerators over the common denominator.

$2 \sqrt{2} (- \sqrt{\frac{\sqrt{2} + x}{\sqrt{2}} (1 - \frac{x}{\sqrt{2}})} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.6

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

$2 \sqrt{2} (- \sqrt{\frac{\sqrt{2} + x}{\sqrt{2}} (\frac{\sqrt{2}}{\sqrt{2}} - \frac{x}{\sqrt{2}})} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.7

Combine the numerators over the common denominator.

$2 \sqrt{2} (- \sqrt{\frac{\sqrt{2} + x}{\sqrt{2}} \cdot \frac{\sqrt{2} - x}{\sqrt{2}}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.8

Multiply $\frac{\sqrt{2} + x}{\sqrt{2}}$ by $\frac{\sqrt{2} - x}{\sqrt{2}}$ .

$2 \sqrt{2} (- \sqrt{\frac{(\sqrt{2} + x) (\sqrt{2} - x)}{\sqrt{2} \sqrt{2}}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.9

Simplify the denominator.

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Step 15.1.9.1

Raise $\sqrt{2}$ to the power of $1$ .

$2 \sqrt{2} (- \sqrt{\frac{(\sqrt{2} + x) (\sqrt{2} - x)}{{\sqrt{2}}^{1} \sqrt{2}}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.9.2

Raise $\sqrt{2}$ to the power of $1$ .

$2 \sqrt{2} (- \sqrt{\frac{(\sqrt{2} + x) (\sqrt{2} - x)}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.9.3

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

$2 \sqrt{2} (- \sqrt{\frac{(\sqrt{2} + x) (\sqrt{2} - x)}{{\sqrt{2}}^{1 + 1}}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.9.4

Add $1$ and $1$ .

$2 \sqrt{2} (- \sqrt{\frac{(\sqrt{2} + x) (\sqrt{2} - x)}{{\sqrt{2}}^{2}}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.10

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 15.1.10.1

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

$2 \sqrt{2} (- \sqrt{\frac{(\sqrt{2} + x) (\sqrt{2} - x)}{{(2^{\frac{1}{2}})}^{2}}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.10.2

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

$2 \sqrt{2} (- \sqrt{\frac{(\sqrt{2} + x) (\sqrt{2} - x)}{2^{\frac{1}{2} \cdot 2}}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.10.3

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

$2 \sqrt{2} (- \sqrt{\frac{(\sqrt{2} + x) (\sqrt{2} - x)}{2^{\frac{2}{2}}}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.10.4

Cancel the common factor of $2$ .

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Step 15.1.10.4.1

Cancel the common factor.

$2 \sqrt{2} (- \sqrt{\frac{(\sqrt{2} + x) (\sqrt{2} - x)}{2^{\frac{2}{2}}}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.10.4.2

Rewrite the expression.

$2 \sqrt{2} (- \sqrt{\frac{(\sqrt{2} + x) (\sqrt{2} - x)}{2^{1}}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.10.5

Evaluate the exponent.

$2 \sqrt{2} (- \sqrt{\frac{(\sqrt{2} + x) (\sqrt{2} - x)}{2}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.11

Expand $(\sqrt{2} + x) (\sqrt{2} - x)$ using the FOIL Method.

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Step 15.1.11.1

Apply the distributive property.

$2 \sqrt{2} (- \sqrt{\frac{\sqrt{2} (\sqrt{2} - x) + x (\sqrt{2} - x)}{2}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.11.2

Apply the distributive property.

$2 \sqrt{2} (- \sqrt{\frac{\sqrt{2} \sqrt{2} + \sqrt{2} (- x) + x (\sqrt{2} - x)}{2}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.11.3

Apply the distributive property.

$2 \sqrt{2} (- \sqrt{\frac{\sqrt{2} \sqrt{2} + \sqrt{2} (- x) + x \sqrt{2} + x (- x)}{2}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.12

Combine the opposite terms in $\sqrt{2} \sqrt{2} + \sqrt{2} (- x) + x \sqrt{2} + x (- x)$ .

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Step 15.1.12.1

Reorder the factors in the terms $\sqrt{2} (- x)$ and $x \sqrt{2}$ .

$2 \sqrt{2} (- \sqrt{\frac{\sqrt{2} \sqrt{2} - x \sqrt{2} + x \sqrt{2} + x (- x)}{2}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.12.2

Add $- x \sqrt{2}$ and $x \sqrt{2}$ .

$2 \sqrt{2} (- \sqrt{\frac{\sqrt{2} \sqrt{2} + 0 + x (- x)}{2}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.12.3

Add $\sqrt{2} \sqrt{2}$ and $0$ .

$2 \sqrt{2} (- \sqrt{\frac{\sqrt{2} \sqrt{2} + x (- x)}{2}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.13

Simplify each term.

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Step 15.1.13.1

Combine using the product rule for radicals.

$2 \sqrt{2} (- \sqrt{\frac{\sqrt{2 \cdot 2} + x (- x)}{2}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.13.2

Multiply $2$ by $2$ .

$2 \sqrt{2} (- \sqrt{\frac{\sqrt{4} + x (- x)}{2}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.13.3

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

$2 \sqrt{2} (- \sqrt{\frac{\sqrt{2^{2}} + x (- x)}{2}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.13.4

Pull terms out from under the radical, assuming positive real numbers.

$2 \sqrt{2} (- \sqrt{\frac{2 + x (- x)}{2}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.13.5

Rewrite using the commutative property of multiplication.

$2 \sqrt{2} (- \sqrt{\frac{2 - x \cdot x}{2}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.13.6

Multiply $x$ by $x$ by adding the exponents.

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Step 15.1.13.6.1

Move $x$ .

$2 \sqrt{2} (- \sqrt{\frac{2 - (x \cdot x)}{2}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.13.6.2

Multiply $x$ by $x$ .

$2 \sqrt{2} (- \sqrt{\frac{2 - x^{2}}{2}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.14

Rewrite $\sqrt{\frac{2 - x^{2}}{2}}$ as $\frac{\sqrt{2 - x^{2}}}{\sqrt{2}}$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{\cos^{3} (\arcsin (\frac{x}{\sqrt{2}}))}{3}) + C$

Step 15.1.15

Simplify the numerator.

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Step 15.1.15.1

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{1 - {(\frac{x}{\sqrt{2}})}^{2}}}^{3}}{3}) + C$

Step 15.1.15.2

Rewrite $1$ as $1^{2}$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{1^{2} - {(\frac{x}{\sqrt{2}})}^{2}}}^{3}}{3}) + C$

Step 15.1.15.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \frac{x}{\sqrt{2}}$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{(1 + \frac{x}{\sqrt{2}}) (1 - \frac{x}{\sqrt{2}})}}^{3}}{3}) + C$

Step 15.1.15.4

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

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{(\frac{\sqrt{2}}{\sqrt{2}} + \frac{x}{\sqrt{2}}) (1 - \frac{x}{\sqrt{2}})}}^{3}}{3}) + C$

Step 15.1.15.5

Combine the numerators over the common denominator.

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{\sqrt{2} + x}{\sqrt{2}} (1 - \frac{x}{\sqrt{2}})}}^{3}}{3}) + C$

Step 15.1.15.6

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

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{\sqrt{2} + x}{\sqrt{2}} (\frac{\sqrt{2}}{\sqrt{2}} - \frac{x}{\sqrt{2}})}}^{3}}{3}) + C$

Step 15.1.15.7

Combine the numerators over the common denominator.

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{\sqrt{2} + x}{\sqrt{2}} \cdot \frac{\sqrt{2} - x}{\sqrt{2}}}}^{3}}{3}) + C$

Step 15.1.15.8

Multiply $\frac{\sqrt{2} + x}{\sqrt{2}}$ by $\frac{\sqrt{2} - x}{\sqrt{2}}$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{(\sqrt{2} + x) (\sqrt{2} - x)}{\sqrt{2} \sqrt{2}}}}^{3}}{3}) + C$

Step 15.1.15.9

Simplify the denominator.

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Step 15.1.15.9.1

Raise $\sqrt{2}$ to the power of $1$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{(\sqrt{2} + x) (\sqrt{2} - x)}{{\sqrt{2}}^{1} \sqrt{2}}}}^{3}}{3}) + C$

Step 15.1.15.9.2

Raise $\sqrt{2}$ to the power of $1$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{(\sqrt{2} + x) (\sqrt{2} - x)}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}}}^{3}}{3}) + C$

Step 15.1.15.9.3

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

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{(\sqrt{2} + x) (\sqrt{2} - x)}{{\sqrt{2}}^{1 + 1}}}}^{3}}{3}) + C$

Step 15.1.15.9.4

Add $1$ and $1$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{(\sqrt{2} + x) (\sqrt{2} - x)}{{\sqrt{2}}^{2}}}}^{3}}{3}) + C$

Step 15.1.15.10

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 15.1.15.10.1

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

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{(\sqrt{2} + x) (\sqrt{2} - x)}{{(2^{\frac{1}{2}})}^{2}}}}^{3}}{3}) + C$

Step 15.1.15.10.2

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

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{(\sqrt{2} + x) (\sqrt{2} - x)}{2^{\frac{1}{2} \cdot 2}}}}^{3}}{3}) + C$

Step 15.1.15.10.3

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

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{(\sqrt{2} + x) (\sqrt{2} - x)}{2^{\frac{2}{2}}}}}^{3}}{3}) + C$

Step 15.1.15.10.4

Cancel the common factor of $2$ .

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Step 15.1.15.10.4.1

Cancel the common factor.

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{(\sqrt{2} + x) (\sqrt{2} - x)}{2^{\frac{2}{2}}}}}^{3}}{3}) + C$

Step 15.1.15.10.4.2

Rewrite the expression.

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{(\sqrt{2} + x) (\sqrt{2} - x)}{2^{1}}}}^{3}}{3}) + C$

Step 15.1.15.10.5

Evaluate the exponent.

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{(\sqrt{2} + x) (\sqrt{2} - x)}{2}}}^{3}}{3}) + C$

Step 15.1.15.11

Expand $(\sqrt{2} + x) (\sqrt{2} - x)$ using the FOIL Method.

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Step 15.1.15.11.1

Apply the distributive property.

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{\sqrt{2} (\sqrt{2} - x) + x (\sqrt{2} - x)}{2}}}^{3}}{3}) + C$

Step 15.1.15.11.2

Apply the distributive property.

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{\sqrt{2} \sqrt{2} + \sqrt{2} (- x) + x (\sqrt{2} - x)}{2}}}^{3}}{3}) + C$

Step 15.1.15.11.3

Apply the distributive property.

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{\sqrt{2} \sqrt{2} + \sqrt{2} (- x) + x \sqrt{2} + x (- x)}{2}}}^{3}}{3}) + C$

Step 15.1.15.12

Combine the opposite terms in $\sqrt{2} \sqrt{2} + \sqrt{2} (- x) + x \sqrt{2} + x (- x)$ .

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Step 15.1.15.12.1

Reorder the factors in the terms $\sqrt{2} (- x)$ and $x \sqrt{2}$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{\sqrt{2} \sqrt{2} - x \sqrt{2} + x \sqrt{2} + x (- x)}{2}}}^{3}}{3}) + C$

Step 15.1.15.12.2

Add $- x \sqrt{2}$ and $x \sqrt{2}$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{\sqrt{2} \sqrt{2} + 0 + x (- x)}{2}}}^{3}}{3}) + C$

Step 15.1.15.12.3

Add $\sqrt{2} \sqrt{2}$ and $0$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{\sqrt{2} \sqrt{2} + x (- x)}{2}}}^{3}}{3}) + C$

Step 15.1.15.13

Simplify each term.

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Step 15.1.15.13.1

Combine using the product rule for radicals.

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{\sqrt{2 \cdot 2} + x (- x)}{2}}}^{3}}{3}) + C$

Step 15.1.15.13.2

Multiply $2$ by $2$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{\sqrt{4} + x (- x)}{2}}}^{3}}{3}) + C$

Step 15.1.15.13.3

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

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{\sqrt{2^{2}} + x (- x)}{2}}}^{3}}{3}) + C$

Step 15.1.15.13.4

Pull terms out from under the radical, assuming positive real numbers.

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{2 + x (- x)}{2}}}^{3}}{3}) + C$

Step 15.1.15.13.5

Rewrite using the commutative property of multiplication.

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{2 - x \cdot x}{2}}}^{3}}{3}) + C$

Step 15.1.15.13.6

Multiply $x$ by $x$ by adding the exponents.

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Step 15.1.15.13.6.1

Move $x$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{2 - (x \cdot x)}{2}}}^{3}}{3}) + C$

Step 15.1.15.13.6.2

Multiply $x$ by $x$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{\sqrt{\frac{2 - x^{2}}{2}}}^{3}}{3}) + C$

Step 15.1.15.14

Rewrite $\sqrt{\frac{2 - x^{2}}{2}}$ as $\frac{\sqrt{2 - x^{2}}}{\sqrt{2}}$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{{(\frac{\sqrt{2 - x^{2}}}{\sqrt{2}})}^{3}}{3}) + C$

Step 15.1.15.15

Apply the product rule to $\frac{\sqrt{2 - x^{2}}}{\sqrt{2}}$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{\frac{{\sqrt{2 - x^{2}}}^{3}}{{\sqrt{2}}^{3}}}{3}) + C$

Step 15.1.15.16

Simplify the numerator.

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Step 15.1.15.16.1

Rewrite ${\sqrt{2 - x^{2}}}^{3}$ as $\sqrt{{(2 - x^{2})}^{3}}$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{\frac{\sqrt{{(2 - x^{2})}^{3}}}{{\sqrt{2}}^{3}}}{3}) + C$

Step 15.1.15.16.2

Factor out ${(2 - x^{2})}^{2}$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{\frac{\sqrt{{(2 - x^{2})}^{2} (2 - x^{2})}}{{\sqrt{2}}^{3}}}{3}) + C$

Step 15.1.15.16.3

Pull terms out from under the radical.

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{\frac{(2 - x^{2}) \sqrt{2 - x^{2}}}{{\sqrt{2}}^{3}}}{3}) + C$

Step 15.1.15.16.4

Apply the distributive property.

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{\frac{2 \sqrt{2 - x^{2}} - x^{2} \sqrt{2 - x^{2}}}{{\sqrt{2}}^{3}}}{3}) + C$

Step 15.1.15.16.5

Factor $\sqrt{2 - x^{2}}$ out of $2 \sqrt{2 - x^{2}} - x^{2} \sqrt{2 - x^{2}}$ .

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Step 15.1.15.16.5.1

Factor $\sqrt{2 - x^{2}}$ out of $2 \sqrt{2 - x^{2}}$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{\frac{\sqrt{2 - x^{2}} \cdot 2 - x^{2} \sqrt{2 - x^{2}}}{{\sqrt{2}}^{3}}}{3}) + C$

Step 15.1.15.16.5.2

Factor $\sqrt{2 - x^{2}}$ out of $- x^{2} \sqrt{2 - x^{2}}$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{\frac{\sqrt{2 - x^{2}} \cdot 2 + \sqrt{2 - x^{2}} (- x^{2})}{{\sqrt{2}}^{3}}}{3}) + C$

Step 15.1.15.16.5.3

Factor $\sqrt{2 - x^{2}}$ out of $\sqrt{2 - x^{2}} \cdot 2 + \sqrt{2 - x^{2}} (- x^{2})$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{\frac{\sqrt{2 - x^{2}} (2 - x^{2})}{{\sqrt{2}}^{3}}}{3}) + C$

Step 15.1.15.17

Simplify the denominator.

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Step 15.1.15.17.1

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{\frac{\sqrt{2 - x^{2}} (2 - x^{2})}{\sqrt{2^{3}}}}{3}) + C$

Step 15.1.15.17.2

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

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{\frac{\sqrt{2 - x^{2}} (2 - x^{2})}{\sqrt{8}}}{3}) + C$

Step 15.1.15.17.3

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 15.1.15.17.3.1

Factor $4$ out of $8$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{\frac{\sqrt{2 - x^{2}} (2 - x^{2})}{\sqrt{4 (2)}}}{3}) + C$

Step 15.1.15.17.3.2

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

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{\frac{\sqrt{2 - x^{2}} (2 - x^{2})}{\sqrt{2^{2} \cdot 2}}}{3}) + C$

Step 15.1.15.17.4

Pull terms out from under the radical.

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{\frac{\sqrt{2 - x^{2}} (2 - x^{2})}{2 \sqrt{2}}}{3}) + C$

Step 15.1.16

Multiply the numerator by the reciprocal of the denominator.

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{\sqrt{2 - x^{2}} (2 - x^{2})}{2 \sqrt{2}} \cdot \frac{1}{3}) + C$

Step 15.1.17

Combine.

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{\sqrt{2 - x^{2}} (2 - x^{2}) \cdot 1}{2 \sqrt{2} \cdot 3}) + C$

Step 15.1.18

Multiply $\sqrt{2 - x^{2}}$ by $1$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{\sqrt{2 - x^{2}} (2 - x^{2})}{2 \sqrt{2} \cdot 3}) + C$

Step 15.1.19

Multiply $3$ by $2$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} + \frac{\sqrt{2 - x^{2}} (2 - x^{2})}{6 \sqrt{2}}) + C$

Step 15.2

To write $- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}}}{\sqrt{2}} \cdot \frac{6}{6} + \frac{\sqrt{2 - x^{2}} (2 - x^{2})}{6 \sqrt{2}}) + C$

Step 15.3

Write each expression with a common denominator of $\sqrt{2} \cdot 6$ , by multiplying each by an appropriate factor of $1$ .

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Step 15.3.1

Multiply $\frac{\sqrt{2 - x^{2}}}{\sqrt{2}}$ by $\frac{6}{6}$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}} \cdot 6}{\sqrt{2} \cdot 6} + \frac{\sqrt{2 - x^{2}} (2 - x^{2})}{6 \sqrt{2}}) + C$

Step 15.3.2

Reorder the factors of $\sqrt{2} \cdot 6$ .

$2 \sqrt{2} (- \frac{\sqrt{2 - x^{2}} \cdot 6}{6 \sqrt{2}} + \frac{\sqrt{2 - x^{2}} (2 - x^{2})}{6 \sqrt{2}}) + C$

Step 15.4

Combine the numerators over the common denominator.

$2 \sqrt{2} \frac{- \sqrt{2 - x^{2}} \cdot 6 + \sqrt{2 - x^{2}} (2 - x^{2})}{6 \sqrt{2}} + C$

Step 15.5

Cancel the common factor of $2 \sqrt{2}$ .

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Step 15.5.1

Factor $2 \sqrt{2}$ out of $6 \sqrt{2}$ .

$2 \sqrt{2} \frac{- \sqrt{2 - x^{2}} \cdot 6 + \sqrt{2 - x^{2}} (2 - x^{2})}{2 \sqrt{2} (3)} + C$

Step 15.5.2

Cancel the common factor.

$2 \sqrt{2} \frac{- \sqrt{2 - x^{2}} \cdot 6 + \sqrt{2 - x^{2}} (2 - x^{2})}{2 \sqrt{2} \cdot 3} + C$

Step 15.5.3

Rewrite the expression.

$\frac{- \sqrt{2 - x^{2}} \cdot 6 + \sqrt{2 - x^{2}} (2 - x^{2})}{3} + C$

Step 15.6

Simplify the numerator.

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Step 15.6.1

Factor $\sqrt{2 - x^{2}}$ out of $- \sqrt{2 - x^{2}} \cdot 6 + \sqrt{2 - x^{2}} (2 - x^{2})$ .

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Step 15.6.1.1

Factor $\sqrt{2 - x^{2}}$ out of $- \sqrt{2 - x^{2}} \cdot 6$ .

$\frac{\sqrt{2 - x^{2}} (- 1 \cdot 6) + \sqrt{2 - x^{2}} (2 - x^{2})}{3} + C$

Step 15.6.1.2

Factor $\sqrt{2 - x^{2}}$ out of $\sqrt{2 - x^{2}} (- 1 \cdot 6) + \sqrt{2 - x^{2}} (2 - x^{2})$ .

$\frac{\sqrt{2 - x^{2}} (- 1 \cdot 6 + 2 - x^{2})}{3} + C$

Step 15.6.2

Multiply $- 1$ by $6$ .

$\frac{\sqrt{2 - x^{2}} (- 6 + 2 - x^{2})}{3} + C$

Step 15.6.3

Add $- 6$ and $2$ .

$\frac{\sqrt{2 - x^{2}} (- 4 - x^{2})}{3} + C$

Step 15.7

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{\sqrt{2 - x^{2}} (- 1 (4) - x^{2})}{3} + C$

Step 15.8

Factor $- 1$ out of $- x^{2}$ .

$\frac{\sqrt{2 - x^{2}} (- 1 (4) - (x^{2}))}{3} + C$

Step 15.9

Factor $- 1$ out of $- 1 (4) - (x^{2})$ .

$\frac{\sqrt{2 - x^{2}} (- 1 (4 + x^{2}))}{3} + C$

Step 15.10

Move the negative in front of the fraction.

$- \frac{\sqrt{2 - x^{2}} (4 + x^{2})}{3} + C$

Step 16

The answer is the antiderivative of the function $f (x) = \frac{x^{3}}{\sqrt{2 - x^{2}}}$ .

$F (x) =$ $- \frac{\sqrt{2 - x^{2}} (4 + x^{2})}{3} + C$