Calculus Examples

$\int \frac{\cos (x)}{\sqrt{9 - \sin^{2} (x)}} d x$

Step 1

Let $u = \sin (x)$ . Then $d u = \cos (x) d x$ , so $\frac{1}{\cos (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

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

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Step 1.1.1

Differentiate $\sin (x)$ .

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

Step 1.1.2

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

$\cos (x)$

Step 1.2

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

$\int \frac{1}{\sqrt{9 - u^{2}}} d u$

Step 2

Rewrite $9$ as $3^{2}$ .

$\int \frac{1}{\sqrt{3^{2} - u^{2}}} d u$

Step 3

The integral of $\frac{1}{\sqrt{3^{2} - u^{2}}}$ with respect to $u$ is $\arcsin (\frac{u}{3})$

$\arcsin (\frac{u}{3}) + C$

Step 4

Rewrite $\arcsin (\frac{u}{3}) + C$ as $\arcsin (\frac{1}{3} u) + C$ .

$\arcsin (\frac{1}{3} u) + C$

Step 5

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

$\arcsin (\frac{1}{3} \sin (x)) + C$