Calculus Examples

$\int_{0}^{\frac{1}{6}} \frac{1}{\sqrt{1 - 9 x^{2}}} d x$

Step 1

Write the expression using exponents.

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Step 1.1

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

$\int_{0}^{\frac{1}{6}} \frac{1}{\sqrt{1^{2} - 9 x^{2}}} d x$

Step 1.2

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$\int_{0}^{\frac{1}{6}} \frac{1}{\sqrt{1^{2} - {(3 x)}^{2}}} d x$

Step 2

Let $u = 3 x$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 3 x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 2.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$3 \cdot 1$

Step 2.1.4

Multiply $3$ by $1$ .

$3$

Step 2.2

Substitute the lower limit in for $x$ in $u = 3 x$ .

$u_{lower} = 3 \cdot 0$

Step 2.3

Multiply $3$ by $0$ .

$u_{lower} = 0$

Step 2.4

Substitute the upper limit in for $x$ in $u = 3 x$ .

$u_{upper} = 3 (\frac{1}{6})$

Step 2.5

Cancel the common factor of $3$ .

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Step 2.5.1

Factor $3$ out of $6$ .

$u_{upper} = 3 \frac{1}{3 (2)}$

Step 2.5.2

Cancel the common factor.

$u_{upper} = 3 \frac{1}{3 \cdot 2}$

Step 2.5.3

Rewrite the expression.

$u_{upper} = \frac{1}{2}$

Step 2.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = \frac{1}{2}$

Step 2.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

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

Step 3

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

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

Step 4

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

$\frac{1}{3} \arcsin (u)]_{0}^{\frac{1}{2}}$

Step 5

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

$\frac{1}{3} (\arcsin (\frac{1}{2}) - \arcsin (0))$

Step 6

Simplify.

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Step 6.1

Simplify each term.

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Step 6.1.1

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

$\frac{1}{3} (\frac{π}{6} - \arcsin (0))$

Step 6.1.2

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

$\frac{1}{3} (\frac{π}{6} - 0)$

Step 6.1.3

Multiply $- 1$ by $0$ .

$\frac{1}{3} (\frac{π}{6} + 0)$

Step 6.2

Add $\frac{π}{6}$ and $0$ .

$\frac{1}{3} \cdot \frac{π}{6}$

Step 6.3

Multiply $\frac{1}{3} \cdot \frac{π}{6}$ .

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Step 6.3.1

Multiply $\frac{1}{3}$ by $\frac{π}{6}$ .

$\frac{π}{3 \cdot 6}$

Step 6.3.2

Multiply $3$ by $6$ .

$\frac{π}{18}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{18}$

Decimal Form:

$0.17453292 \dots$

Step 8