Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

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

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

Step 3.2

Simplify.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

$\frac{π}{6} - \arcsin (0)$

Step 3.2.1.2

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

$\frac{π}{6} - 0$

Step 3.2.1.3

Multiply $- 1$ by $0$ .

$\frac{π}{6} + 0$

Step 3.2.2

Add $\frac{π}{6}$ and $0$ .

$\frac{π}{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{6}$

Decimal Form:

$0.52359877 \dots$

Step 5