Calculus Examples

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

Step 1

Let $x = \frac{1}{2} \tan (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \frac{\sec^{2} (t)}{2} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\frac{\sec^{2} (t)}{2}$ is positive.

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

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{4 {(\frac{1}{2} \tan (t))}^{2} + 1}$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Combine $\frac{1}{2}$ and $\tan (t)$ .

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

Step 2.1.1.2

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

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

Step 2.1.1.3

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

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

Step 2.1.1.4

Cancel the common factor of $4$ .

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Step 2.1.1.4.1

Cancel the common factor.

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

Step 2.1.1.4.2

Rewrite the expression.

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

Step 2.1.2

Apply pythagorean identity.

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

Step 2.1.3

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

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

Step 2.2

Simplify.

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Step 2.2.1

Combine $\frac{1}{2}$ and $\tan (t)$ .

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

Step 2.2.2

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

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

Step 2.2.3

Multiply by the reciprocal of the fraction to divide by $\frac{\tan (t) \sec (t)}{2}$ .

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

Step 2.2.4

Multiply $\frac{2}{\tan (t) \sec (t)}$ by $1$ .

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

Step 2.2.5

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

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

Step 2.2.6

Move $2$ to the left of $\tan (t) \sec (t)$ .

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

Step 2.2.7

Cancel the common factor of $2$ .

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Step 2.2.7.1

Cancel the common factor.

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

Step 2.2.7.2

Rewrite the expression.

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

Step 2.2.8

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

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Step 2.2.8.1

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

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

Step 2.2.8.2

Cancel the common factors.

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Step 2.2.8.2.1

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

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

Step 2.2.8.2.2

Cancel the common factor.

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

Step 2.2.8.2.3

Rewrite the expression.

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

Step 2.2.9

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

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

Step 2.2.10

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

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

Step 2.2.11

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

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

Step 2.2.12

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

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Step 2.2.12.1

Cancel the common factor.

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

Step 2.2.12.2

Rewrite the expression.

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

Step 2.2.13

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

$\int \csc (t) d t$

Step 3

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

$\ln (| \csc (t) - \cot (t) |) + C$

Step 4

Replace all occurrences of $t$ with $\arctan (2 x)$ .

$\ln (| \csc (\arctan (2 x)) - \cot (\arctan (2 x)) |) + C$