Calculus Examples

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

Step 1

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

$\int \frac{1}{\frac{2}{3} \tan (t) \sqrt{9 {(\frac{2}{3} \tan (t))}^{2} + 4}} \cdot \frac{2 \sec^{2} (t)}{3} d t$

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

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

$\int \frac{1}{\frac{2}{3} \tan (t) \sqrt{9 {(\frac{2 \tan (t)}{3})}^{2} + 4}} \cdot \frac{2 \sec^{2} (t)}{3} d t$

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 $\frac{2 \tan (t)}{3}$ .

$\int \frac{1}{\frac{2}{3} \tan (t) \sqrt{9 \frac{{(2 \tan (t))}^{2}}{3^{2}} + 4}} \cdot \frac{2 \sec^{2} (t)}{3} d t$

Step 2.1.1.2.2

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

$\int \frac{1}{\frac{2}{3} \tan (t) \sqrt{9 \frac{2^{2} \tan^{2} (t)}{3^{2}} + 4}} \cdot \frac{2 \sec^{2} (t)}{3} d t$

Step 2.1.1.3

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

$\int \frac{1}{\frac{2}{3} \tan (t) \sqrt{9 \frac{4 \tan^{2} (t)}{3^{2}} + 4}} \cdot \frac{2 \sec^{2} (t)}{3} d t$

Step 2.1.1.4

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

$\int \frac{1}{\frac{2}{3} \tan (t) \sqrt{9 \frac{4 \tan^{2} (t)}{9} + 4}} \cdot \frac{2 \sec^{2} (t)}{3} d t$

Step 2.1.1.5

Cancel the common factor of $9$ .

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Step 2.1.1.5.1

Cancel the common factor.

$\int \frac{1}{\frac{2}{3} \tan (t) \sqrt{9 \frac{4 \tan^{2} (t)}{9} + 4}} \cdot \frac{2 \sec^{2} (t)}{3} d t$

Step 2.1.1.5.2

Rewrite the expression.

$\int \frac{1}{\frac{2}{3} \tan (t) \sqrt{4 \tan^{2} (t) + 4}} \cdot \frac{2 \sec^{2} (t)}{3} d t$

Step 2.1.2

Factor $4$ out of $4 \tan^{2} (t) + 4$ .

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Step 2.1.2.1

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

$\int \frac{1}{\frac{2}{3} \tan (t) \sqrt{4 (\tan^{2} (t)) + 4}} \cdot \frac{2 \sec^{2} (t)}{3} d t$

Step 2.1.2.2

Factor $4$ out of $4$ .

$\int \frac{1}{\frac{2}{3} \tan (t) \sqrt{4 (\tan^{2} (t)) + 4 (1)}} \cdot \frac{2 \sec^{2} (t)}{3} d t$

Step 2.1.2.3

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

$\int \frac{1}{\frac{2}{3} \tan (t) \sqrt{4 (\tan^{2} (t) + 1)}} \cdot \frac{2 \sec^{2} (t)}{3} d t$

Step 2.1.3

Apply pythagorean identity.

$\int \frac{1}{\frac{2}{3} \tan (t) \sqrt{4 \sec^{2} (t)}} \cdot \frac{2 \sec^{2} (t)}{3} d t$

Step 2.1.4

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

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

Step 2.1.5

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

$\int \frac{1}{\frac{2}{3} \tan (t) (2 \sec (t))} \cdot \frac{2 \sec^{2} (t)}{3} d t$

Step 2.2

Simplify.

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Step 2.2.1

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

$\int \frac{1}{\frac{2 \tan (t)}{3} (2 \sec (t))} \cdot \frac{2 \sec^{2} (t)}{3} d t$

Step 2.2.2

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

$\int \frac{1}{\frac{2 (2 \tan (t))}{3} \sec (t)} \cdot \frac{2 \sec^{2} (t)}{3} d t$

Step 2.2.3

Multiply $2$ by $2$ .

$\int \frac{1}{\frac{4 \tan (t)}{3} \sec (t)} \cdot \frac{2 \sec^{2} (t)}{3} d t$

Step 2.2.4

Combine $\frac{4 \tan (t)}{3}$ and $\sec (t)$ .

$\int \frac{1}{\frac{4 \tan (t) \sec (t)}{3}} \cdot \frac{2 \sec^{2} (t)}{3} d t$

Step 2.2.5

Multiply by the reciprocal of the fraction to divide by $\frac{4 \tan (t) \sec (t)}{3}$ .

$\int 1 \frac{3}{4 \tan (t) \sec (t)} \frac{2 \sec^{2} (t)}{3} d t$

Step 2.2.6

Multiply $\frac{3}{4 \tan (t) \sec (t)}$ by $1$ .

$\int \frac{3}{4 \tan (t) \sec (t)} \cdot \frac{2 \sec^{2} (t)}{3} d t$

Step 2.2.7

Multiply $\frac{3}{4 \tan (t) \sec (t)}$ by $\frac{2 \sec^{2} (t)}{3}$ .

$\int \frac{3 (2 \sec^{2} (t))}{4 \tan (t) \sec (t) \cdot 3} d t$

Step 2.2.8

Multiply $2$ by $3$ .

$\int \frac{6 \sec^{2} (t)}{4 \tan (t) \sec (t) \cdot 3} d t$

Step 2.2.9

Multiply $3$ by $4$ .

$\int \frac{6 \sec^{2} (t)}{12 \tan (t) \sec (t)} d t$

Step 2.2.10

Cancel the common factor of $6$ and $12$ .

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Step 2.2.10.1

Factor $6$ out of $6 \sec^{2} (t)$ .

$\int \frac{6 (\sec^{2} (t))}{12 \tan (t) \sec (t)} d t$

Step 2.2.10.2

Cancel the common factors.

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Step 2.2.10.2.1

Factor $6$ out of $12 \tan (t) \sec (t)$ .

$\int \frac{6 (\sec^{2} (t))}{6 (2 \tan (t) \sec (t))} d t$

Step 2.2.10.2.2

Cancel the common factor.

$\int \frac{6 \sec^{2} (t)}{6 (2 \tan (t) \sec (t))} d t$

Step 2.2.10.2.3

Rewrite the expression.

$\int \frac{\sec^{2} (t)}{2 \tan (t) \sec (t)} d t$

Step 2.2.11

Cancel the common factor of $\sec^{2} (t)$ and $\sec (t)$ .

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Step 2.2.11.1

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

$\int \frac{\sec (t) \sec (t)}{2 \tan (t) \sec (t)} d t$

Step 2.2.11.2

Cancel the common factors.

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Step 2.2.11.2.1

Factor $\sec (t)$ out of $2 \tan (t) \sec (t)$ .

$\int \frac{\sec (t) \sec (t)}{\sec (t) (2 \tan (t))} d t$

Step 2.2.11.2.2

Cancel the common factor.

$\int \frac{\sec (t) \sec (t)}{\sec (t) (2 \tan (t))} d t$

Step 2.2.11.2.3

Rewrite the expression.

$\int \frac{\sec (t)}{2 \tan (t)} d t$

Step 3

Since $\frac{1}{2}$ is constant with respect to $t$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int \frac{\sec (t)}{\tan (t)} d t$

Step 4

Simplify.

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Step 4.1

Rewrite $\tan (t)$ in terms of sines and cosines.

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

Step 4.2

Rewrite $\sec (t)$ in terms of sines and cosines.

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

Step 4.3

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

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

Step 4.4

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

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Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

Rewrite the expression.

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

Step 4.5

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

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

Step 5

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

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

Step 6

Simplify.

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

Step 7

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

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

Step 8

Reorder terms.

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