Calculus Examples

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

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.

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

Step 2

Simplify terms.

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Step 2.1

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

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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 $2 \tan (t)$ .

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

Step 2.1.1.2

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

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

Step 2.1.2

Factor $4$ out of $4$ .

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

Rearrange terms.

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

Step 2.1.6

Apply pythagorean identity.

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

Step 2.1.7

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

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

Step 2.1.8

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

$\int \sec (t) d t$

Step 3

The integral of $\sec (t)$ with respect to $t$ is $\ln (| \sec (t) + \tan (t) |)$ .

$\ln (| \sec (t) + \tan (t) |) + C$

Step 4

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

$\ln (| \sec (\arctan (\frac{x}{2})) + \tan (\arctan (\frac{x}{2})) |) + C$

Step 5

Reorder terms.

$\ln (| \sec (\arctan (\frac{1}{2} x)) + \tan (\arctan (\frac{1}{2} x)) |) + C$