Calculus Examples

$\int \frac{\sin (x)}{\sqrt{4 + \cos^{2} (x)}} d x$

Step 1

Let $u = \cos (x)$ . Then $d u = - \sin (x) d x$ , so $- d u = \sin (x) d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = \cos (x)$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $\cos (x)$ .

$\frac{d}{d x} [\cos (x)]$

Step 1.1.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$- \sin (x)$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

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

Step 2

Move the negative in front of the fraction.

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

Step 3

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

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

Step 4

Let $u = 2 \tan (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d u = 2 \sec^{2} (t) d t$ . 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 5

Simplify terms.

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Step 5.1

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

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

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 5.1.1.2

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

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

Step 5.1.2

Factor $4$ out of $4$ .

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

Step 5.1.3

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 5.1.4

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 5.1.5

Rearrange terms.

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

Step 5.1.6

Apply pythagorean identity.

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

Step 5.1.7

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 5.1.8

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

$- \int \sec (t) d t$

Step 6

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

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

Step 7

Simplify.

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

Step 8

Substitute back in for each integration substitution variable.

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Step 8.1

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

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

Step 8.2

Replace all occurrences of $u$ with $\cos (x)$ .

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

Step 9

Reorder terms.

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