Calculus Examples

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

Step 1

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

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Step 1.1

Raise $x$ to the power of $1$ .

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

Step 1.2

Factor $x$ out of $x^{1}$ .

$\int \frac{1}{\sqrt{x \cdot 1 - x^{2}}} d x$

Step 1.3

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

$\int \frac{1}{\sqrt{x \cdot 1 + x (- x)}} d x$

Step 1.4

Factor $x$ out of $x \cdot 1 + x (- x)$ .

$\int \frac{1}{\sqrt{x (1 - x)}} d x$

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.

$x \cdot 1 + x (- x)$

Step 2.1.2

Multiply $x$ by $1$ .

$x + x (- x)$

Step 2.1.3

Rewrite using the commutative property of multiplication.

$x - x \cdot x$

Step 2.1.4

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

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Step 2.1.4.1

Move $x$ .

$x - (x \cdot x)$

Step 2.1.4.2

Multiply $x$ by $x$ .

$x - x^{2}$

Step 2.1.5

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

$- x^{2} + x$

Step 2.2

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

$a = - 1$

$b = 1$

$c = 0$

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 $d = \frac{b}{2 a}$ .

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

Step 2.4.2

Cancel the common factor of $1$ and $- 1$ .

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Step 2.4.2.1

Rewrite $1$ as $- 1 (- 1)$ .

$d = \frac{- 1 (- 1)}{2 \cdot - 1}$

Step 2.4.2.2

Move the negative in front of the fraction.

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

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 $e = c - \frac{b^{2}}{4 a}$ .

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

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

One to any power is one.

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

Step 2.5.2.1.2

Multiply $4$ by $- 1$ .

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

Step 2.5.2.1.3

Move the negative in front of the fraction.

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

Step 2.5.2.1.4

Multiply $- - \frac{1}{4}$ .

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Step 2.5.2.1.4.1

Multiply $- 1$ by $- 1$ .

$e = 0 + 1 (\frac{1}{4})$

Step 2.5.2.1.4.2

Multiply $\frac{1}{4}$ by $1$ .

$e = 0 + \frac{1}{4}$

Step 2.5.2.2

Add $0$ and $\frac{1}{4}$ .

$e = \frac{1}{4}$

Step 2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x - \frac{1}{2})}^{2} + \frac{1}{4}$ .

$\int \frac{1}{\sqrt{- {(x - \frac{1}{2})}^{2} + \frac{1}{4}}} d x$

Step 3

Let $u = x - \frac{1}{2}$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = x - \frac{1}{2}$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $x - \frac{1}{2}$ .

$\frac{d}{d x} [x - \frac{1}{2}]$

Step 3.1.2

By the Sum Rule, the derivative of $x - \frac{1}{2}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{1}{2}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{1}{2}]$

Step 3.1.3

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

$1 + \frac{d}{d x} [- \frac{1}{2}]$

Step 3.1.4

Since $- \frac{1}{2}$ is constant with respect to $x$ , the derivative of $- \frac{1}{2}$ with respect to $x$ is $0$ .

$1 + 0$

Step 3.1.5

Add $1$ and $0$ .

$1$

Step 3.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{1}{\sqrt{- u^{2} + \frac{1}{4}}} d u$

Step 4

Write the expression using exponents.

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Step 4.1

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

$\int \frac{1}{\sqrt{- u^{2} + \frac{1^{2}}{4}}} d u$

Step 4.2

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

$\int \frac{1}{\sqrt{- u^{2} + \frac{1^{2}}{2^{2}}}} d u$

Step 5

Rewrite $\frac{1^{2}}{2^{2}}$ as ${(\frac{1}{2})}^{2}$ .

$\int \frac{1}{\sqrt{- u^{2} + {(\frac{1}{2})}^{2}}} d u$

Step 6

Reorder $- u^{2}$ and ${(\frac{1}{2})}^{2}$ .

$\int \frac{1}{\sqrt{{(\frac{1}{2})}^{2} - u^{2}}} d u$

Step 7

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

$\arcsin (\frac{u}{\frac{1}{2}}) + C$

Step 8

Simplify.

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Step 8.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

$\arcsin (u \cdot 2) + C$

Step 8.2

Move $2$ to the left of $u$ .

$\arcsin (2 u) + C$

Step 9

Replace all occurrences of $u$ with $x - \frac{1}{2}$ .

$\arcsin (2 (x - \frac{1}{2})) + C$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$\arcsin (2 x + 2 (- \frac{1}{2})) + C$

Step 10.2

Cancel the common factor of $2$ .

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Step 10.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$\arcsin (2 x + 2 (\frac{- 1}{2})) + C$

Step 10.2.2

Cancel the common factor.

$\arcsin (2 x + 2 (\frac{- 1}{2})) + C$

Step 10.2.3

Rewrite the expression.

$\arcsin (2 x - 1) + C$