Calculus Examples

$\int \frac{1}{x \sqrt{9 - 4 x^{2}}} d x$

Step 1

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

$\int \frac{1}{\frac{3}{2} \sin (t) \sqrt{9 - 4 {(\frac{3}{2} \sin (t))}^{2}}} \cdot \frac{3 \cos (t)}{2} d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{9 - 4 {(\frac{3}{2} \sin (t))}^{2}}$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Combine $\frac{3}{2}$ and $\sin (t)$ .

$\int \frac{1}{\frac{3}{2} \sin (t) \sqrt{9 - 4 {(\frac{3 \sin (t)}{2})}^{2}}} \cdot \frac{3 \cos (t)}{2} d t$

Step 2.1.1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.1.2.1

Apply the product rule to $\frac{3 \sin (t)}{2}$ .

$\int \frac{1}{\frac{3}{2} \sin (t) \sqrt{9 - 4 \frac{{(3 \sin (t))}^{2}}{2^{2}}}} \cdot \frac{3 \cos (t)}{2} d t$

Step 2.1.1.2.2

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

$\int \frac{1}{\frac{3}{2} \sin (t) \sqrt{9 - 4 \frac{3^{2} \sin^{2} (t)}{2^{2}}}} \cdot \frac{3 \cos (t)}{2} d t$

Step 2.1.1.3

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

$\int \frac{1}{\frac{3}{2} \sin (t) \sqrt{9 - 4 \frac{9 \sin^{2} (t)}{2^{2}}}} \cdot \frac{3 \cos (t)}{2} d t$

Step 2.1.1.4

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

$\int \frac{1}{\frac{3}{2} \sin (t) \sqrt{9 - 4 \frac{9 \sin^{2} (t)}{4}}} \cdot \frac{3 \cos (t)}{2} d t$

Step 2.1.1.5

Cancel the common factor of $4$ .

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Step 2.1.1.5.1

Factor $4$ out of $- 4$ .

$\int \frac{1}{\frac{3}{2} \sin (t) \sqrt{9 + 4 (- 1) \frac{9 \sin^{2} (t)}{4}}} \cdot \frac{3 \cos (t)}{2} d t$

Step 2.1.1.5.2

Cancel the common factor.

$\int \frac{1}{\frac{3}{2} \sin (t) \sqrt{9 + 4 \cdot - 1 \frac{9 \sin^{2} (t)}{4}}} \cdot \frac{3 \cos (t)}{2} d t$

Step 2.1.1.5.3

Rewrite the expression.

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

Step 2.1.1.6

Multiply $9$ by $- 1$ .

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

Step 2.1.2

Factor $9$ out of $9$ .

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

Apply pythagorean identity.

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

Step 2.1.6

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

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

Step 2.1.7

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

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

Step 2.2

Simplify.

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Step 2.2.1

Combine $\frac{3}{2}$ and $\sin (t)$ .

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

Step 2.2.2

Combine $3$ and $\frac{3 \sin (t)}{2}$ .

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

Step 2.2.3

Multiply $3$ by $3$ .

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

Step 2.2.4

Combine $\frac{9 \sin (t)}{2}$ and $\cos (t)$ .

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

Step 2.2.5

Multiply by the reciprocal of the fraction to divide by $\frac{9 \sin (t) \cos (t)}{2}$ .

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

Step 2.2.6

Multiply $\frac{2}{9 \sin (t) \cos (t)}$ by $1$ .

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

Step 2.2.7

Multiply $\frac{2}{9 \sin (t) \cos (t)}$ by $\frac{3 \cos (t)}{2}$ .

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

Step 2.2.8

Multiply $3$ by $2$ .

$\int \frac{6 \cos (t)}{9 \sin (t) \cos (t) \cdot 2} d t$

Step 2.2.9

Multiply $2$ by $9$ .

$\int \frac{6 \cos (t)}{18 \sin (t) \cos (t)} d t$

Step 2.2.10

Cancel the common factor of $6$ and $18$ .

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Step 2.2.10.1

Factor $6$ out of $6 \cos (t)$ .

$\int \frac{6 (\cos (t))}{18 \sin (t) \cos (t)} d t$

Step 2.2.10.2

Cancel the common factors.

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Step 2.2.10.2.1

Factor $6$ out of $18 \sin (t) \cos (t)$ .

$\int \frac{6 (\cos (t))}{6 (3 \sin (t) \cos (t))} d t$

Step 2.2.10.2.2

Cancel the common factor.

$\int \frac{6 \cos (t)}{6 (3 \sin (t) \cos (t))} d t$

Step 2.2.10.2.3

Rewrite the expression.

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

Step 2.2.11

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

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Step 2.2.11.1

Cancel the common factor.

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

Step 2.2.11.2

Rewrite the expression.

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

Step 3

Since $\frac{1}{3}$ is constant with respect to $t$ , move $\frac{1}{3}$ out of the integral.

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

Step 4

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

$\frac{1}{3} \int \csc (t) d t$

Step 5

The integral of $\csc (t)$ with respect to $t$ is $\ln (| \csc (t) - \cot (t) |)$ .

$\frac{1}{3} (\ln (| \csc (t) - \cot (t) |) + C)$

Step 6

Simplify.

$\frac{1}{3} \ln (| \csc (t) - \cot (t) |) + C$

Step 7

Replace all occurrences of $t$ with $\arcsin (\frac{2 x}{3})$ .

$\frac{1}{3} \ln (| \csc (\arcsin (\frac{2 x}{3})) - \cot (\arcsin (\frac{2 x}{3})) |) + C$

Step 8

Reorder terms.

$\frac{1}{3} \ln (| \csc (\arcsin (\frac{2}{3} x)) - \cot (\arcsin (\frac{2}{3} x)) |) + C$