Calculus Examples

$\int_{0}^{\frac{1}{2}} \frac{1}{\sqrt{1 - x^{2}}} d x d x$

Step 1

Rewrite $1$ as $1^{2}$ .

$\int_{0}^{\frac{1}{2}} \frac{1}{\sqrt{1^{2} - x^{2}}} d x d x$

Step 2

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

$\arcsin (x)]_{0}^{\frac{1}{2}} d x$

Step 3

Evaluate $\arcsin (x)$ at $\frac{1}{2}$ and at $0$ .

$(\arcsin (\frac{1}{2}) - \arcsin (0)) d x$

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

The exact value of $\arcsin (\frac{1}{2})$ is $\frac{π}{6}$ .

$(\frac{π}{6} - \arcsin (0)) d x$

Step 4.1.2

The exact value of $\arcsin (0)$ is $0$ .

$(\frac{π}{6} - 0) d x$

Step 4.1.3

Multiply $- 1$ by $0$ .

$(\frac{π}{6} + 0) d x$

Step 4.2

Add $\frac{π}{6}$ and $0$ .

$\frac{π}{6} d x$

Step 4.3

Combine $\frac{π}{6}$ and $d$ .

$\frac{π d}{6} x$

Step 4.4

Combine $\frac{π d}{6}$ and $x$ .

$\frac{π d x}{6}$

Step 5