Calculus Examples

$\int \frac{1}{x^{2} \sqrt{9 - x^{2}}} d x d x$

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.

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

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 $3 \sin (t)$ .

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

Step 2.1.1.2

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

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

Step 2.1.1.3

Multiply $9$ by $- 1$ .

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

Step 2.1.2

Factor $9$ out of $9$ .

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

Apply pythagorean identity.

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

Step 2.1.6

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

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

Step 2.1.7

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

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

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

Factor $3 \cos (t)$ out of ${(3 \sin (t))}^{2} (3 \cos (t))$ .

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

Step 2.2.1.2

Cancel the common factor.

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

Step 2.2.1.3

Rewrite the expression.

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

Step 2.2.2

Simplify.

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

$\int \frac{1}{9 \sin^{2} (t)} d t d x$

Step 3

Since $\frac{1}{9}$ is constant with respect to $t$ , move $\frac{1}{9}$ out of the integral.

$\frac{1}{9} \int \frac{1}{\sin^{2} (t)} d t d x$

Step 4

Simplify.

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Step 4.1

Convert from $\frac{1}{\sin^{2} (t)}$ to $\csc^{2} (t)$ .

$\frac{1}{9} \int \csc^{2} (t) d t d x$

Step 4.2

Combine $d$ and $\frac{1}{9}$ .

$\frac{d}{9} \int \csc^{2} (t) d t x$

Step 4.3

Combine $x$ and $\frac{d}{9}$ .

$\frac{x d}{9} \int \csc^{2} (t) d t$

Step 5

Since the derivative of $- \cot (t)$ is $\csc^{2} (t)$ , the integral of $\csc^{2} (t)$ is $- \cot (t)$ .

$\frac{x d}{9} (- \cot (t) + C)$

Step 6

Simplify.

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Step 6.1

Rewrite $\frac{x d}{9} (- \cot (t) + C)$ as $\frac{1}{9} x d (- \cot (t)) + C$ .

$\frac{1}{9} x d (- \cot (t)) + C$

Step 6.2

Simplify.

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Step 6.2.1

Combine $\frac{1}{9}$ and $x$ .

$\frac{x}{9} d (- \cot (t)) + C$

Step 6.2.2

Combine $\frac{x}{9}$ and $d$ .

$\frac{x d}{9} (- \cot (t)) + C$

Step 6.2.3

Combine $\frac{x d}{9}$ and $\cot (t)$ .

$- \frac{x d \cot (t)}{9} + C$

Step 7

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

$- \frac{x d \cot (\arcsin (\frac{x}{3}))}{9} + C$

Step 8

Simplify the numerator.

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Step 8.1

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, $\cot (\arcsin (\frac{x}{3}))$ is $\frac{\sqrt{1 - {(\frac{x}{3})}^{2}}}{\frac{x}{3}}$ .

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

Step 8.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.3

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

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

Step 8.4

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

$- \frac{x d (\sqrt{(1 + \frac{x}{3}) (1 - \frac{x}{3})} \frac{3}{x})}{9} + C$

Step 8.5

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

$- \frac{x d (\sqrt{(\frac{3}{3} + \frac{x}{3}) (1 - \frac{x}{3})} \frac{3}{x})}{9} + C$

Step 8.6

Combine the numerators over the common denominator.

$- \frac{x d (\sqrt{\frac{3 + x}{3} (1 - \frac{x}{3})} \frac{3}{x})}{9} + C$

Step 8.7

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

$- \frac{x d (\sqrt{\frac{3 + x}{3} (\frac{3}{3} - \frac{x}{3})} \frac{3}{x})}{9} + C$

Step 8.8

Combine the numerators over the common denominator.

$- \frac{x d (\sqrt{\frac{3 + x}{3} \cdot \frac{3 - x}{3}} \frac{3}{x})}{9} + C$

Step 8.9

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

$- \frac{x d (\sqrt{\frac{(3 + x) (3 - x)}{3 \cdot 3}} \frac{3}{x})}{9} + C$

Step 8.10

Multiply $3$ by $3$ .

$- \frac{x d (\sqrt{\frac{(3 + x) (3 - x)}{9}} \frac{3}{x})}{9} + C$

Step 8.11

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

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Step 8.11.1

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

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

Step 8.11.2

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

$- \frac{x d (\sqrt{\frac{1^{2} ((3 + x) (3 - x))}{3^{2} \cdot 1}} \frac{3}{x})}{9} + C$

Step 8.11.3

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

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

Step 8.12

Pull terms out from under the radical.

$- \frac{x d (\frac{1}{3} \sqrt{(3 + x) (3 - x)} \frac{3}{x})}{9} + C$

Step 8.13

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

$- \frac{x d (\frac{\sqrt{(3 + x) (3 - x)}}{3} \cdot \frac{3}{x})}{9} + C$

Step 8.14

Cancel the common factor of $3$ .

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Step 8.14.1

Cancel the common factor.

$- \frac{x d (\frac{\sqrt{(3 + x) (3 - x)}}{3} \cdot \frac{3}{x})}{9} + C$

Step 8.14.2

Rewrite the expression.

$- \frac{x d (\sqrt{(3 + x) (3 - x)} \frac{1}{x})}{9} + C$

Step 8.15

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

$- \frac{x d \frac{\sqrt{(3 + x) (3 - x)}}{x}}{9} + C$

Step 8.16

Combine exponents.

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Step 8.16.1

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

$- \frac{d \frac{x \sqrt{(3 + x) (3 - x)}}{x}}{9} + C$

Step 8.16.2

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

$- \frac{\frac{d (x \sqrt{(3 + x) (3 - x)})}{x}}{9} + C$

Step 8.17

Remove unnecessary parentheses.

$- \frac{\frac{d x \sqrt{(3 + x) (3 - x)}}{x}}{9} + C$

Step 8.18

Reduce the expression by cancelling the common factors.

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Step 8.18.1

Reduce the expression $\frac{d x \sqrt{(3 + x) (3 - x)}}{x}$ by cancelling the common factors.

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Step 8.18.1.1

Cancel the common factor.

$- \frac{\frac{d x \sqrt{(3 + x) (3 - x)}}{x}}{9} + C$

Step 8.18.1.2

Rewrite the expression.

$- \frac{\frac{d \sqrt{(3 + x) (3 - x)}}{1}}{9} + C$

Step 8.18.2

Divide $d \sqrt{(3 + x) (3 - x)}$ by $1$ .

$- \frac{d \sqrt{(3 + x) (3 - x)}}{9} + C$

$- \frac{1}{9} d \sqrt{(3 + x) (3 - x)} + C$