Calculus Examples

$\int \arcsin (\cos (x)) d x$

Step 1

Let $u_{2} = \arcsin (\cos (x))$ . Then $d u_{2} = - d x$ , so $- d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 1.1

Let $u_{2} = \arcsin (\cos (x))$ . Find $\frac{d u_{2}}{d x}$ .

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Step 1.1.1

Differentiate $\arcsin (\cos (x))$ .

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

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arcsin (x)$ and $g (x) = \cos (x)$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u_{1}$ as $\cos (x)$ .

$\frac{d}{d u_{1}} [\arcsin (u_{1})] \frac{d}{d x} [\cos (x)]$

Step 1.1.2.2

The derivative of $\arcsin (u_{1})$ with respect to $u_{1}$ is $\frac{1}{\sqrt{1 - {u_{1}}^{2}}}$ .

$\frac{1}{\sqrt{1 - {u_{1}}^{2}}} \frac{d}{d x} [\cos (x)]$

Step 1.1.2.3

Replace all occurrences of $u_{1}$ with $\cos (x)$ .

$\frac{1}{\sqrt{1 - \cos^{2} (x)}} \frac{d}{d x} [\cos (x)]$

Step 1.1.3

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

$\frac{1}{\sqrt{1 - \cos^{2} (x)}} (- \sin (x))$

Step 1.1.4

Combine $\frac{1}{\sqrt{1 - \cos^{2} (x)}}$ and $\sin (x)$ .

$- \frac{\sin (x)}{\sqrt{1 - \cos^{2} (x)}}$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Apply pythagorean identity.

$- \frac{\sin (x)}{\sqrt{\sin^{2} (x)}}$

Step 1.1.5.2

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

$- \frac{\sin (x)}{\sin (x)}$

Step 1.1.5.3

Cancel the common factor of $\sin (x)$ .

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Step 1.1.5.3.1

Cancel the common factor.

$- \frac{\sin (x)}{\sin (x)}$

Step 1.1.5.3.2

Rewrite the expression.

$- 1 \cdot 1$

Step 1.1.5.4

Multiply $- 1$ by $1$ .

$- 1$

Step 1.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\int - u_{2} d u_{2}$

Step 2

Since $- 1$ is constant with respect to $u_{2}$ , move $- 1$ out of the integral.

$- \int u_{2} d u_{2}$

Step 3

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

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

Step 4

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

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

Step 5

Replace all occurrences of $u_{2}$ with $\arcsin (\cos (x))$ .

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