Calculus Examples

$\frac{1}{\sqrt{1 - x} \sqrt{1 + x}}$

Step 1

Write $\frac{1}{\sqrt{1 - x} \sqrt{1 + x}}$ as a function.

$f (x) = \frac{1}{\sqrt{1 - x} \sqrt{1 + x}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

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

Step 4

Combine using the product rule for radicals.

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

Step 5

Complete the square.

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Step 5.1

Simplify the expression.

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Step 5.1.1

Expand $(1 - x) (1 + x)$ using the FOIL Method.

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Step 5.1.1.1

Apply the distributive property.

$1 (1 + x) - x (1 + x)$

Step 5.1.1.2

Apply the distributive property.

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

Step 5.1.1.3

Apply the distributive property.

$1 \cdot 1 + 1 x - x \cdot 1 - x \cdot x$

Step 5.1.2

Simplify and combine like terms.

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Step 5.1.2.1

Simplify each term.

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Step 5.1.2.1.1

Multiply $1$ by $1$ .

$1 + 1 x - x \cdot 1 - x \cdot x$

Step 5.1.2.1.2

Multiply $x$ by $1$ .

$1 + x - x \cdot 1 - x \cdot x$

Step 5.1.2.1.3

Multiply $- 1$ by $1$ .

$1 + x - x - x \cdot x$

Step 5.1.2.1.4

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

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Step 5.1.2.1.4.1

Move $x$ .

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

Step 5.1.2.1.4.2

Multiply $x$ by $x$ .

$1 + x - x - x^{2}$

Step 5.1.2.2

Subtract $x$ from $x$ .

$1 + 0 - x^{2}$

Step 5.1.2.3

Add $1$ and $0$ .

$1 - x^{2}$

Step 5.1.3

Reorder $1$ and $- x^{2}$ .

$- x^{2} + 1$

Step 5.2

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

$a = - 1$

$b = 0$

$c = 1$

Step 5.3

Consider the vertex form of a parabola.

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

Step 5.4

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

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Step 5.4.1

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

$d = \frac{0}{2 \cdot - 1}$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

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

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Step 5.4.2.1.1

Factor $2$ out of $0$ .

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

Step 5.4.2.1.2

Move the negative one from the denominator of $\frac{0}{- 1}$ .

$d = - 1 \cdot 0$

Step 5.4.2.2

Rewrite $- 1 \cdot 0$ as $- 0$ .

$d = - 0$

Step 5.4.2.3

Multiply $- 1$ by $0$ .

$d = 0$

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

Simplify the right side.

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Step 5.5.2.1

Simplify each term.

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Step 5.5.2.1.1

Raising $0$ to any positive power yields $0$ .

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

Step 5.5.2.1.2

Multiply $4$ by $- 1$ .

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

Step 5.5.2.1.3

Divide $0$ by $- 4$ .

$e = 1 - 0$

Step 5.5.2.1.4

Multiply $- 1$ by $0$ .

$e = 1 + 0$

Step 5.5.2.2

Add $1$ and $0$ .

$e = 1$

Step 5.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x + 0)}^{2} + 1$ .

$\int \frac{1}{\sqrt{- {(x + 0)}^{2} + 1}} d x$

Step 6

Let $u = x + 0$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.1

Let $u = x + 0$ . Find $\frac{d u}{d x}$ .

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Step 6.1.1

Differentiate $x + 0$ .

$\frac{d}{d x} [x + 0]$

Step 6.1.2

By the Sum Rule, the derivative of $x + 0$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [0]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [0]$

Step 6.1.3

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

$1 + \frac{d}{d x} [0]$

Step 6.1.4

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

$1 + 0$

Step 6.1.5

Add $1$ and $0$ .

$1$

Step 6.2

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

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

Step 7

Simplify the expression.

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Step 7.1

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

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

Step 7.2

Reorder $- u^{2}$ and $1^{2}$ .

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

Step 8

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

$\arcsin (u) + C$

Step 9

Replace all occurrences of $u$ with $x + 0$ .

$\arcsin (x + 0) + C$

Step 10

Add $x$ and $0$ .

$\arcsin (x) + C$

Step 11

The answer is the antiderivative of the function $f (x) = \frac{1}{\sqrt{1 - x} \sqrt{1 + x}}$ .

$F (x) =$ $\arcsin (x) + C$