Calculus Examples

$\int \frac{x^{3}}{\sqrt{9 - x^{2}}} d x$

Step 1

Let $x = 3 \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = 3 \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $3 \cos (t)$ is positive.

$\int \frac{{(3 \sin (t))}^{3}}{\sqrt{9 - {(3 \sin (t))}^{2}}} (3 \cos (t)) d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{9 - {(3 \sin (t))}^{2}}$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the product rule to $3 \sin (t)$ .

$\int \frac{{(3 \sin (t))}^{3}}{\sqrt{9 - (3^{2} \sin^{2} (t))}} (3 \cos (t)) d t$

Step 2.1.1.2

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

$\int \frac{{(3 \sin (t))}^{3}}{\sqrt{9 - (9 \sin^{2} (t))}} (3 \cos (t)) d t$

Step 2.1.1.3

Multiply $9$ by $- 1$ .

$\int \frac{{(3 \sin (t))}^{3}}{\sqrt{9 - 9 \sin^{2} (t)}} (3 \cos (t)) d t$

Step 2.1.2

Factor $9$ out of $9$ .

$\int \frac{{(3 \sin (t))}^{3}}{\sqrt{9 (1) - 9 \sin^{2} (t)}} (3 \cos (t)) d t$

Step 2.1.3

Factor $9$ out of $- 9 \sin^{2} (t)$ .

$\int \frac{{(3 \sin (t))}^{3}}{\sqrt{9 (1) + 9 (- \sin^{2} (t))}} (3 \cos (t)) d t$

Step 2.1.4

Factor $9$ out of $9 (1) + 9 (- \sin^{2} (t))$ .

$\int \frac{{(3 \sin (t))}^{3}}{\sqrt{9 (1 - \sin^{2} (t))}} (3 \cos (t)) d t$

Step 2.1.5

Apply pythagorean identity.

$\int \frac{{(3 \sin (t))}^{3}}{\sqrt{9 \cos^{2} (t)}} (3 \cos (t)) d t$

Step 2.1.6

Rewrite $9 \cos^{2} (t)$ as ${(3 \cos (t))}^{2}$ .

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

Step 2.1.7

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

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

Step 2.2

Reduce the expression by cancelling the common factors.

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Step 2.2.1

Cancel the common factor of $3 \cos (t)$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Rewrite the expression.

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

Step 2.2.2

Simplify.

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Step 2.2.2.1

Factor $3$ out of $3 \sin (t)$ .

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

Step 2.2.2.2

Apply the product rule to $3 (\sin (t))$ .

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

Step 2.2.2.3

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

$\int 27 \sin^{3} (t) d t$

Step 3

Since $27$ is constant with respect to $t$ , move $27$ out of the integral.

$27 \int \sin^{3} (t) d t$

Step 4

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

$27 \int \sin^{2} (t) \sin (t) d t$

Step 5

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

$27 \int (1 - \cos^{2} (t)) \sin (t) d t$

Step 6

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 6.1

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

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Step 6.1.1

Differentiate $\cos (t)$ .

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

Step 6.1.2

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

$- \sin (t)$

Step 6.2

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

$27 \int - 1 + u^{2} d u$

Step 7

Split the single integral into multiple integrals.

$27 (\int - 1 d u + \int u^{2} d u)$

Step 8

Apply the constant rule.

$27 (- u + C + \int u^{2} d u)$

Step 9

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

$27 (- u + C + \frac{1}{3} u^{3} + C)$

Step 10

Simplify.

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Step 10.1

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

$27 (- u + C + \frac{u^{3}}{3} + C)$

Step 10.2

Simplify.

$27 (- u + \frac{1}{3} u^{3}) + C$

Step 11

Substitute back in for each integration substitution variable.

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Step 11.1

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

$27 (- \cos (t) + \frac{1}{3} \cos^{3} (t)) + C$

Step 11.2

Replace all occurrences of $t$ with $\arcsin (\frac{x}{3})$ .

$27 (- \cos (\arcsin (\frac{x}{3})) + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{3}))) + C$

Step 12

Simplify.

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Step 12.1

Simplify each term.

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Step 12.1.1

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{x}{3})}^{2}}, \frac{x}{3})$ , $(\sqrt{1^{2} - {(\frac{x}{3})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{x}{3})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{x}{3})}^{2}}, \frac{x}{3})$ . Therefore, $\cos (\arcsin (\frac{x}{3}))$ is $\sqrt{1 - {(\frac{x}{3})}^{2}}$ .

$27 (- \sqrt{1 - {(\frac{x}{3})}^{2}} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{3}))) + C$

Step 12.1.2

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

$27 (- \sqrt{1^{2} - {(\frac{x}{3})}^{2}} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{3}))) + C$

Step 12.1.3

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 $b = \frac{x}{3}$ .

$27 (- \sqrt{(1 + \frac{x}{3}) (1 - \frac{x}{3})} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{3}))) + C$

Step 12.1.4

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

$27 (- \sqrt{(\frac{3}{3} + \frac{x}{3}) (1 - \frac{x}{3})} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{3}))) + C$

Step 12.1.5

Combine the numerators over the common denominator.

$27 (- \sqrt{\frac{3 + x}{3} (1 - \frac{x}{3})} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{3}))) + C$

Step 12.1.6

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

$27 (- \sqrt{\frac{3 + x}{3} (\frac{3}{3} - \frac{x}{3})} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{3}))) + C$

Step 12.1.7

Combine the numerators over the common denominator.

$27 (- \sqrt{\frac{3 + x}{3} \cdot \frac{3 - x}{3}} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{3}))) + C$

Step 12.1.8

Multiply $\frac{3 + x}{3}$ by $\frac{3 - x}{3}$ .

$27 (- \sqrt{\frac{(3 + x) (3 - x)}{3 \cdot 3}} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{3}))) + C$

Step 12.1.9

Multiply $3$ by $3$ .

$27 (- \sqrt{\frac{(3 + x) (3 - x)}{9}} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{3}))) + C$

Step 12.1.10

Rewrite $\frac{(3 + x) (3 - x)}{9}$ as ${(\frac{1}{3})}^{2} ((3 + x) (3 - x))$ .

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Step 12.1.10.1

Factor the perfect power $1^{2}$ out of $(3 + x) (3 - x)$ .

$27 (- \sqrt{\frac{1^{2} ((3 + x) (3 - x))}{9}} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{3}))) + C$

Step 12.1.10.2

Factor the perfect power $3^{2}$ out of $9$ .

$27 (- \sqrt{\frac{1^{2} ((3 + x) (3 - x))}{3^{2} \cdot 1}} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{3}))) + C$

Step 12.1.10.3

Rearrange the fraction $\frac{1^{2} ((3 + x) (3 - x))}{3^{2} \cdot 1}$ .

$27 (- \sqrt{{(\frac{1}{3})}^{2} ((3 + x) (3 - x))} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{3}))) + C$

Step 12.1.11

Pull terms out from under the radical.

$27 (- (\frac{1}{3} \sqrt{(3 + x) (3 - x)}) + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{3}))) + C$

Step 12.1.12

Combine $\frac{1}{3}$ and $\sqrt{(3 + x) (3 - x)}$ .

$27 (- \frac{\sqrt{(3 + x) (3 - x)}}{3} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{3}))) + C$

Step 12.2

Combine $\frac{1}{3}$ and $\cos^{3} (\arcsin (\frac{x}{3}))$ .

$27 (- \frac{\sqrt{(3 + x) (3 - x)}}{3} + \frac{\cos^{3} (\arcsin (\frac{x}{3}))}{3}) + C$

Step 12.3

Apply the distributive property.

$27 (- \frac{\sqrt{(3 + x) (3 - x)}}{3}) + 27 \frac{\cos^{3} (\arcsin (\frac{x}{3}))}{3} + C$

Step 12.4

Cancel the common factor of $3$ .

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Step 12.4.1

Move the leading negative in $- \frac{\sqrt{(3 + x) (3 - x)}}{3}$ into the numerator.

$27 \frac{- \sqrt{(3 + x) (3 - x)}}{3} + 27 \frac{\cos^{3} (\arcsin (\frac{x}{3}))}{3} + C$

Step 12.4.2

Factor $3$ out of $27$ .

$3 (9) \frac{- \sqrt{(3 + x) (3 - x)}}{3} + 27 \frac{\cos^{3} (\arcsin (\frac{x}{3}))}{3} + C$

Step 12.4.3

Cancel the common factor.

$3 \cdot 9 \frac{- \sqrt{(3 + x) (3 - x)}}{3} + 27 \frac{\cos^{3} (\arcsin (\frac{x}{3}))}{3} + C$

Step 12.4.4

Rewrite the expression.

$9 (- \sqrt{(3 + x) (3 - x)}) + 27 \frac{\cos^{3} (\arcsin (\frac{x}{3}))}{3} + C$

Step 12.5

Multiply $- 1$ by $9$ .

$- 9 \sqrt{(3 + x) (3 - x)} + 27 \frac{\cos^{3} (\arcsin (\frac{x}{3}))}{3} + C$

Step 12.6

Cancel the common factor of $3$ .

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Step 12.6.1

Factor $3$ out of $27$ .

$- 9 \sqrt{(3 + x) (3 - x)} + 3 (9) \frac{\cos^{3} (\arcsin (\frac{x}{3}))}{3} + C$

Step 12.6.2

Cancel the common factor.

$- 9 \sqrt{(3 + x) (3 - x)} + 3 \cdot 9 \frac{\cos^{3} (\arcsin (\frac{x}{3}))}{3} + C$

Step 12.6.3

Rewrite the expression.

$- 9 \sqrt{(3 + x) (3 - x)} + 9 \cos^{3} (\arcsin (\frac{x}{3})) + C$

Step 13

Reorder terms.

$- 9 \sqrt{(3 + x) (3 - x)} + 9 \cos^{3} (\arcsin (\frac{1}{3} x)) + C$