Calculus Examples

$\int \frac{\sin (x)}{\sqrt{\cos (x)}} d x$

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 $d u$ .

$\int \frac{- 1}{\sqrt{u}} d u$

Step 2

Move the negative in front of the fraction.

$\int - \frac{1}{\sqrt{u}} d u$

Step 3

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

$- \int \frac{1}{\sqrt{u}} d u$

Step 4

Apply basic rules of exponents.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

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

Step 4.2

Move $u^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 4.3

Multiply the exponents in ${(u^{\frac{1}{2}})}^{- 1}$ .

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Step 4.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.3.2

Combine $\frac{1}{2}$ and $- 1$ .

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

Step 4.3.3

Move the negative in front of the fraction.

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

Step 5

By the Power Rule, the integral of $u^{- \frac{1}{2}}$ with respect to $u$ is $2 u^{\frac{1}{2}}$ .

$- (2 u^{\frac{1}{2}} + C)$

Step 6

Simplify.

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Step 6.1

Rewrite $- (2 u^{\frac{1}{2}} + C)$ as $- 1 \cdot 2 u^{\frac{1}{2}} + C$ .

$- 1 \cdot 2 u^{\frac{1}{2}} + C$

Step 6.2

Multiply $- 1$ by $2$ .

$- 2 u^{\frac{1}{2}} + C$

Step 7

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

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