Calculus Examples

$\int \frac{1}{x^{2} \sqrt{4 - x^{2}}} d x$

Step 1

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

$\int \frac{1}{{(2 \sin (t))}^{2} \sqrt{4 - {(2 \sin (t))}^{2}}} (2 \cos (t)) d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{4 - {(2 \sin (t))}^{2}}$ .

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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 $2 \sin (t)$ .

$\int \frac{1}{{(2 \sin (t))}^{2} \sqrt{4 - (2^{2} \sin^{2} (t))}} (2 \cos (t)) d t$

Step 2.1.1.2

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

$\int \frac{1}{{(2 \sin (t))}^{2} \sqrt{4 - (4 \sin^{2} (t))}} (2 \cos (t)) d t$

Step 2.1.1.3

Multiply $4$ by $- 1$ .

$\int \frac{1}{{(2 \sin (t))}^{2} \sqrt{4 - 4 \sin^{2} (t)}} (2 \cos (t)) d t$

Step 2.1.2

Factor $4$ out of $4$ .

$\int \frac{1}{{(2 \sin (t))}^{2} \sqrt{4 (1) - 4 \sin^{2} (t)}} (2 \cos (t)) d t$

Step 2.1.3

Factor $4$ out of $- 4 \sin^{2} (t)$ .

$\int \frac{1}{{(2 \sin (t))}^{2} \sqrt{4 (1) + 4 (- \sin^{2} (t))}} (2 \cos (t)) d t$

Step 2.1.4

Factor $4$ out of $4 (1) + 4 (- \sin^{2} (t))$ .

$\int \frac{1}{{(2 \sin (t))}^{2} \sqrt{4 (1 - \sin^{2} (t))}} (2 \cos (t)) d t$

Step 2.1.5

Apply pythagorean identity.

$\int \frac{1}{{(2 \sin (t))}^{2} \sqrt{4 \cos^{2} (t)}} (2 \cos (t)) d t$

Step 2.1.6

Rewrite $4 \cos^{2} (t)$ as ${(2 \cos (t))}^{2}$ .

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

Step 2.1.7

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

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

Step 2.2

Reduce the expression by cancelling the common factors.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

Cancel the common factor.

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

Step 2.2.1.3

Rewrite the expression.

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

Step 2.2.2

Simplify.

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Step 2.2.2.1

Factor $2$ out of $2 \sin (t)$ .

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

$\int \frac{1}{4 \sin^{2} (t)} d t$

Step 3

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

$\frac{1}{4} \int \frac{1}{\sin^{2} (t)} d t$

Step 4

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

$\frac{1}{4} \int \csc^{2} (t) d t$

Step 5

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

$\frac{1}{4} (- \cot (t) + C)$

Step 6

Simplify.

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Step 6.1

Simplify.

$\frac{1}{4} (- \cot (t)) + C$

Step 6.2

Combine $\frac{1}{4}$ and $\cot (t)$ .

$- \frac{\cot (t)}{4} + C$

Step 7

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

$- \frac{\cot (\arcsin (\frac{x}{2}))}{4} + C$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

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

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

Step 8.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.1.3

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

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

Step 8.1.4

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 $b = \frac{x}{2}$ .

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

Step 8.1.5

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

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

Step 8.1.6

Combine the numerators over the common denominator.

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

Step 8.1.7

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

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

Step 8.1.8

Combine the numerators over the common denominator.

$- \frac{\sqrt{\frac{2 + x}{2} \cdot \frac{2 - x}{2}} \frac{2}{x}}{4} + C$

Step 8.1.9

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

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

Step 8.1.10

Multiply $2$ by $2$ .

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

Step 8.1.11

Rewrite $\frac{(2 + x) (2 - x)}{4}$ as ${(\frac{1}{2})}^{2} ((2 + x) (2 - x))$ .

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Step 8.1.11.1

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

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

Step 8.1.11.2

Factor the perfect power $2^{2}$ out of $4$ .

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

Step 8.1.11.3

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

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

Step 8.1.12

Pull terms out from under the radical.

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

Step 8.1.13

Combine $\frac{1}{2}$ and $\sqrt{(2 + x) (2 - x)}$ .

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

Step 8.1.14

Cancel the common factor of $2$ .

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Step 8.1.14.1

Cancel the common factor.

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

Step 8.1.14.2

Rewrite the expression.

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

Step 8.1.15

Combine $\sqrt{(2 + x) (2 - x)}$ and $\frac{1}{x}$ .

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

Step 8.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.3

Multiply $\frac{\sqrt{(2 + x) (2 - x)}}{x}$ by $\frac{1}{4}$ .

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

Step 8.4

Move $4$ to the left of $x$ .

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