Calculus Examples

$\int_{0}^{1} \frac{2}{\sqrt{4 - x^{2}}} d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{0}^{1} \frac{1}{\sqrt{4 - x^{2}}} d x$

Step 2

Rewrite $4$ as $2^{2}$ .

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

Step 3

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

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

Step 4

Substitute and simplify.

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Step 4.1

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

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

Step 4.2

Cancel the common factor of $0$ and $2$ .

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Step 4.2.1

Factor $2$ out of $0$ .

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

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $2$ out of $2$ .

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

Step 4.2.2.2

Cancel the common factor.

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

Step 4.2.2.3

Rewrite the expression.

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

Step 4.2.2.4

Divide $0$ by $1$ .

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

Step 5

Simplify.

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Step 5.1

Simplify each term.

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Step 5.1.1

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

$2 (\frac{π}{6} - \arcsin (0))$

Step 5.1.2

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

$2 (\frac{π}{6} - 0)$

Step 5.1.3

Multiply $- 1$ by $0$ .

$2 (\frac{π}{6} + 0)$

Step 5.2

Add $\frac{π}{6}$ and $0$ .

$2 \frac{π}{6}$

Step 5.3

Cancel the common factor of $2$ .

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Step 5.3.1

Factor $2$ out of $6$ .

$2 \frac{π}{2 (3)}$

Step 5.3.2

Cancel the common factor.

$2 \frac{π}{2 \cdot 3}$

Step 5.3.3

Rewrite the expression.

$\frac{π}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{3}$

Decimal Form:

$1.04719755 \dots$

Step 7