Calculus Examples

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

Step 1

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

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

Step 2

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

$\frac{2}{π} \arcsin (x)]_{- 1}^{0}$

Step 3

Evaluate $\arcsin (x)$ at $0$ and at $- 1$ .

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

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 $\arcsin (0)$ is $0$ .

$\frac{2}{π} (0 - \arcsin (- 1))$

Step 4.1.2

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

$\frac{2}{π} (0 - - \frac{π}{2})$

Step 4.1.3

Multiply $- - \frac{π}{2}$ .

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Step 4.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{2}{π} (0 + 1 \frac{π}{2})$

Step 4.1.3.2

Multiply $\frac{π}{2}$ by $1$ .

$\frac{2}{π} (0 + \frac{π}{2})$

Step 4.2

Add $0$ and $\frac{π}{2}$ .

$\frac{2}{π} \cdot \frac{π}{2}$

Step 4.3

Cancel the common factor of $2$ .

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Step 4.3.1

Cancel the common factor.

$\frac{2}{π} \cdot \frac{π}{2}$

Step 4.3.2

Rewrite the expression.

$\frac{1}{π} π$

Step 4.4

Cancel the common factor of $π$ .

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Step 4.4.1

Cancel the common factor.

$\frac{1}{π} π$

Step 4.4.2

Rewrite the expression.

$1$

Step 5