Calculus Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

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

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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 $2 \tan (t)$ .

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

Step 2.1.1.2

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

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

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

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

Step 2.1.2.2

Factor $4$ out of $4$ .

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

Step 2.1.2.3

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

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

Step 2.1.3

Apply pythagorean identity.

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

Step 2.1.4

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

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

Step 2.1.5

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

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

Step 2.2

Simplify terms.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

Cancel the common factor.

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

Step 2.2.1.4

Rewrite the expression.

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

Step 2.2.2

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

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

Step 2.2.3

Simplify.

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Step 2.2.3.1

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

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

Step 2.2.3.2

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

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

Step 2.2.3.3

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Rewrite $\sec (x)$ as $\frac{\csc (x)}{\cot (x)}$ .

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

Step 4.2

Apply the reciprocal identity.

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

Step 4.3

Simplify.

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Step 4.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.2

Combine.

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

Step 4.3.3

Multiply $\csc (t)$ by $1$ .

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

Step 4.3.4

Simplify the denominator.

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Step 4.3.4.1

Apply the product rule to $\frac{1}{\cot (t)}$ .

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

Step 4.3.4.2

One to any power is one.

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

Step 4.3.5

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

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

Step 4.3.6

Reduce the expression $\frac{\cot (t)}{{\cot (t)}^{2}}$ by cancelling the common factors.

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Step 4.3.6.1

Multiply by $1$ .

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

Step 4.3.6.2

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

$\frac{1}{4} \int \frac{\csc (t)}{\frac{\cot (t) \cdot 1}{\cot (t) \cot (t)}} d t$

Step 4.3.6.3

Cancel the common factor.

$\frac{1}{4} \int \frac{\csc (t)}{\frac{\cot (t) \cdot 1}{\cot (t) \cot (t)}} d t$

Step 4.3.6.4

Rewrite the expression.

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

Step 4.3.7

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Simplify.

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

Step 6.2

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

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

Step 7

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

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

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 $(1, \frac{x}{2})$ , $(1, 0)$ , and the origin. Then $\arctan (\frac{x}{2})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \frac{x}{2})$ . Therefore, $\csc (\arctan (\frac{x}{2}))$ 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

Apply the product rule to $\frac{x}{2}$ .

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

Step 8.1.4

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

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

Step 8.1.5

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

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

Step 8.1.6

Combine the numerators over the common denominator.

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

Step 8.1.7

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

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Step 8.1.7.1

Factor the perfect power $1^{2}$ out of $4 + x^{2}$ .

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

Step 8.1.7.2

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

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

Step 8.1.7.3

Rearrange the fraction $\frac{1^{2} (4 + x^{2})}{2^{2} \cdot 1}$ .

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

Step 8.1.8

Pull terms out from under the radical.

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

Step 8.1.9

Combine $\frac{1}{2}$ and $\sqrt{4 + x^{2}}$ .

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

Step 8.1.10

Combine.

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

Step 8.1.11

Cancel the common factor of $2$ .

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Step 8.1.11.1

Cancel the common factor.

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

Step 8.1.11.2

Rewrite the expression.

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

Step 8.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.3

Combine.

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

Step 8.4

Multiply $\sqrt{4 + x^{2}}$ by $1$ .

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

Step 8.5

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

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