Calculus Examples

$\int_{0}^{3} \frac{1}{\sqrt{9 - x^{2}}} d x$

Step 1

Rewrite $9$ as $3^{2}$ .

$\int_{0}^{3} \frac{1}{\sqrt{3^{2} - x^{2}}} d x$

Step 2

The integral of $\frac{1}{\sqrt{3^{2} - x^{2}}}$ with respect to $x$ is $\arcsin (\frac{x}{3})$

$\arcsin (\frac{x}{3})]_{0}^{3}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $\arcsin (\frac{x}{3})$ at $3$ and at $0$ .

$(\arcsin (\frac{3}{3})) - \arcsin (\frac{0}{3})$

Step 3.2

Simplify.

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Step 3.2.1

Cancel the common factor of $3$ .

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Step 3.2.1.1

Cancel the common factor.

$\arcsin (\frac{3}{3}) - \arcsin (\frac{0}{3})$

Step 3.2.1.2

Rewrite the expression.

$\arcsin (1) - \arcsin (\frac{0}{3})$

Step 3.2.2

Cancel the common factor of $0$ and $3$ .

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Step 3.2.2.1

Factor $3$ out of $0$ .

$\arcsin (1) - \arcsin (\frac{3 (0)}{3})$

Step 3.2.2.2

Cancel the common factors.

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Step 3.2.2.2.1

Factor $3$ out of $3$ .

$\arcsin (1) - \arcsin (\frac{3 \cdot 0}{3 \cdot 1})$

Step 3.2.2.2.2

Cancel the common factor.

$\arcsin (1) - \arcsin (\frac{3 \cdot 0}{3 \cdot 1})$

Step 3.2.2.2.3

Rewrite the expression.

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

Step 3.2.2.2.4

Divide $0$ by $1$ .

$\arcsin (1) - \arcsin (0)$

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 $\arcsin (1)$ is $\frac{π}{2}$ .

$\frac{π}{2} - \arcsin (0)$

Step 4.1.2

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

$\frac{π}{2} - 0$

Step 4.1.3

Multiply $- 1$ by $0$ .

$\frac{π}{2} + 0$

Step 4.2

Add $\frac{π}{2}$ and $0$ .

$\frac{π}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{2}$

Decimal Form:

$1.57079632 \dots$

Step 6