Calculus Examples

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

Step 1

Factor $x$ out of $6 x - x^{2}$ .

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Step 1.1

Factor $x$ out of $6 x$ .

$\int_{3}^{6} \frac{1}{\sqrt{x \cdot 6 - x^{2}}} d x$

Step 1.2

Factor $x$ out of $- x^{2}$ .

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

Step 1.3

Factor $x$ out of $x \cdot 6 + x (- x)$ .

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

Step 2

Complete the square.

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Step 2.1

Simplify the expression.

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Step 2.1.1

Apply the distributive property.

$x \cdot 6 + x (- x)$

Step 2.1.2

Move $6$ to the left of $x$ .

$6 \cdot x + x (- x)$

Step 2.1.3

Rewrite using the commutative property of multiplication.

$6 \cdot x - x \cdot x$

Step 2.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.4.1

Move $x$ .

$6 x - (x \cdot x)$

Step 2.1.4.2

Multiply $x$ by $x$ .

$6 x - x^{2}$

Step 2.1.5

Reorder $6 x$ and $- x^{2}$ .

$- x^{2} + 6 x$

Step 2.2

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 1$

$b = 6$

$c = 0$

Step 2.3

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 2.4

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 2.4.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{6}{2 \cdot - 1}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Cancel the common factor of $6$ and $2$ .

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Step 2.4.2.1.1

Factor $2$ out of $6$ .

$d = \frac{2 \cdot 3}{2 \cdot - 1}$

Step 2.4.2.1.2

Move the negative one from the denominator of $\frac{3}{- 1}$ .

$d = - 1 \cdot 3$

Step 2.4.2.2

Multiply $- 1$ by $3$ .

$d = - 3$

Step 2.5

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 2.5.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{6^{2}}{4 \cdot - 1}$

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Simplify each term.

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Step 2.5.2.1.1

Raise $6$ to the power of $2$ .

$e = 0 - \frac{36}{4 \cdot - 1}$

Step 2.5.2.1.2

Multiply $4$ by $- 1$ .

$e = 0 - \frac{36}{- 4}$

Step 2.5.2.1.3

Divide $36$ by $- 4$ .

$e = 0 - - 9$

Step 2.5.2.1.4

Multiply $- 1$ by $- 9$ .

$e = 0 + 9$

Step 2.5.2.2

Add $0$ and $9$ .

$e = 9$

Step 2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x - 3)}^{2} + 9$ .

$\int_{3}^{6} \frac{1}{\sqrt{- {(x - 3)}^{2} + 9}} d x$

Step 3

Let $u = x - 3$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

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

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Step 3.1.1

Differentiate $x - 3$ .

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

Step 3.1.2

By the Sum Rule, the derivative of $x - 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

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

Step 3.1.3

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

$1 + \frac{d}{d x} [- 3]$

Step 3.1.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$1 + 0$

Step 3.1.5

Add $1$ and $0$ .

$1$

Step 3.2

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

$u_{lower} = 3 - 3$

Step 3.3

Subtract $3$ from $3$ .

$u_{lower} = 0$

Step 3.4

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

$u_{upper} = 6 - 3$

Step 3.5

Subtract $3$ from $6$ .

$u_{upper} = 3$

Step 3.6

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

$u_{lower} = 0$

$u_{upper} = 3$

Step 3.7

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

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

Step 4

Simplify the expression.

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Step 4.1

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

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

Step 4.2

Reorder $- u^{2}$ and $3^{2}$ .

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

Step 5

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

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

Step 6

Substitute and simplify.

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Step 6.1

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

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

Step 6.2

Simplify.

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Step 6.2.1

Cancel the common factor of $3$ .

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Rewrite the expression.

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

Step 6.2.2

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

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Step 6.2.2.1

Factor $3$ out of $0$ .

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

Step 6.2.2.2

Cancel the common factors.

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Step 6.2.2.2.1

Factor $3$ out of $3$ .

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

Step 6.2.2.2.2

Cancel the common factor.

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

Step 6.2.2.2.3

Rewrite the expression.

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

Step 6.2.2.2.4

Divide $0$ by $1$ .

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

Step 7

Simplify.

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Step 7.1

Simplify each term.

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Step 7.1.1

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

$\frac{π}{2} - \arcsin (0)$

Step 7.1.2

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

$\frac{π}{2} - 0$

Step 7.1.3

Multiply $- 1$ by $0$ .

$\frac{π}{2} + 0$

Step 7.2

Add $\frac{π}{2}$ and $0$ .

$\frac{π}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{2}$

Decimal Form:

$1.57079632 \dots$

Step 9