Calculus Examples

\int \frac{8}{\sqrt{12 - x^{2} - 4 x}} d x

$\int \frac{8}{\sqrt{12 - x^{2} - 4 x}} d x$

Step 1

Factor by grouping.

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Step 1.1

Reorder terms.

\int \frac{8}{\sqrt{- x^{2} - 4 x + 12}} d x

$\int \frac{8}{\sqrt{- x^{2} - 4 x + 12}} d x$

Step 1.2

For a polynomial of the form

a x^{2} + b x + c

$a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is

a \cdot c = - 1 \cdot 12 = - 12

$a \cdot c = - 1 \cdot 12 = - 12$ and whose sum is

b = - 4

$b = - 4$ .

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Step 1.2.1

Factor

- 4

$- 4$ out of

- 4 x

$- 4 x$ .

\int \frac{8}{\sqrt{- x^{2} - 4 (x) + 12}} d x

$\int \frac{8}{\sqrt{- x^{2} - 4 (x) + 12}} d x$

Step 1.2.2

Rewrite

- 4

$- 4$ as

2

$2$ plus

- 6

$- 6$

\int \frac{8}{\sqrt{- x^{2} + (2 - 6) x + 12}} d x

$\int \frac{8}{\sqrt{- x^{2} + (2 - 6) x + 12}} d x$

Step 1.2.3

Apply the distributive property.

\int \frac{8}{\sqrt{- x^{2} + 2 x - 6 x + 12}} d x

$\int \frac{8}{\sqrt{- x^{2} + 2 x - 6 x + 12}} d x$

\int \frac{8}{\sqrt{- x^{2} + 2 x - 6 x + 12}} d x

$\int \frac{8}{\sqrt{- x^{2} + 2 x - 6 x + 12}} d x$

Step 1.3

Factor out the greatest common factor from each group.

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Step 1.3.1

Group the first two terms and the last two terms.

\int \frac{8}{\sqrt{(- x^{2} + 2 x) - 6 x + 12}} d x

$\int \frac{8}{\sqrt{(- x^{2} + 2 x) - 6 x + 12}} d x$

Step 1.3.2

Factor out the greatest common factor (GCF) from each group.

\int \frac{8}{\sqrt{x (- x + 2) + 6 (- x + 2)}} d x

$\int \frac{8}{\sqrt{x (- x + 2) + 6 (- x + 2)}} d x$

\int \frac{8}{\sqrt{x (- x + 2) + 6 (- x + 2)}} d x

$\int \frac{8}{\sqrt{x (- x + 2) + 6 (- x + 2)}} d x$

Step 1.4

Factor the polynomial by factoring out the greatest common factor,

- x + 2

$- x + 2$ .

\int \frac{8}{\sqrt{(- x + 2) (x + 6)}} d x

$\int \frac{8}{\sqrt{(- x + 2) (x + 6)}} d x$

\int \frac{8}{\sqrt{(- x + 2) (x + 6)}} d x

$\int \frac{8}{\sqrt{(- x + 2) (x + 6)}} d x$

Step 2

Since

8

$8$ is constant with respect to

x

$x$ , move

8

$8$ out of the integral.

8 \int \frac{1}{\sqrt{(- x + 2) (x + 6)}} d x

$8 \int \frac{1}{\sqrt{(- x + 2) (x + 6)}} d x$

Step 3

Complete the square.

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Step 3.1

Simplify the expression.

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Step 3.1.1

Expand

(- x + 2) (x + 6)

$(- x + 2) (x + 6)$ using the FOIL Method.

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Step 3.1.1.1

Apply the distributive property.

$- x (x + 6) + 2 (x + 6)$

Step 3.1.1.2

Apply the distributive property.

$- x \cdot x - x \cdot 6 + 2 (x + 6)$

Step 3.1.1.3

Apply the distributive property.

$- x \cdot x - x \cdot 6 + 2 x + 2 \cdot 6$

Step 3.1.2

Simplify and combine like terms.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Multiply

$x$ by

$x$ by adding the exponents.

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Step 3.1.2.1.1.1

Move

$x$ .

$- (x \cdot x) - x \cdot 6 + 2 x + 2 \cdot 6$

Step 3.1.2.1.1.2

Multiply

$x$ by

$x$ .

$- x^{2} - x \cdot 6 + 2 x + 2 \cdot 6$

Step 3.1.2.1.2

Multiply

$6$ by

$- 1$ .

$- x^{2} - 6 x + 2 x + 2 \cdot 6$

Step 3.1.2.1.3

Multiply

$2$ by

$6$ .

$- x^{2} - 6 x + 2 x + 12$

Step 3.1.2.2

Add

$- 6 x$ and

$2 x$ .

$- x^{2} - 4 x + 12$

Step 3.2

Use the form

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

$a$ ,

$b$ , and

$c$ .

$a = - 1$

$b = - 4$

$c = 12$

Step 3.3

Consider the vertex form of a parabola.

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

Step 3.4

Find the value of

$d$ using the formula

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

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Step 3.4.1

Substitute the values of

$a$ and

$b$ into the formula

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

$d = \frac{- 4}{2 \cdot - 1}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Cancel the common factor of

$- 4$ and

$2$ .

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Step 3.4.2.1.1

Factor

$2$ out of

$- 4$ .

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

Step 3.4.2.1.2

Move the negative one from the denominator of

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

$d = - 1 \cdot - 2$

Step 3.4.2.2

Rewrite

$- 1 \cdot - 2$ as

$- - 2$ .

$d = - - 2$

Step 3.4.2.3

Multiply

$- 1$ by

$- 2$ .

$d = 2$

Step 3.5

Find the value of

$e$ using the formula

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

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Step 3.5.1

Substitute the values of

$c$ ,

$b$ and

$a$ into the formula

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

$e = 12 - \frac{{(- 4)}^{2}}{4 \cdot - 1}$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify each term.

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Step 3.5.2.1.1

Cancel the common factor of

${(- 4)}^{2}$ and

$4$ .

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Step 3.5.2.1.1.1

Rewrite

$- 4$ as

$- 1 (4)$ .

$e = 12 - \frac{{(- 1 (4))}^{2}}{4 \cdot - 1}$

Step 3.5.2.1.1.2

Apply the product rule to

$- 1 (4)$ .

$e = 12 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 \cdot - 1}$

Step 3.5.2.1.1.3

Raise

$- 1$ to the power of

$2$ .

$e = 12 - \frac{1 \cdot 4^{2}}{4 \cdot - 1}$

Step 3.5.2.1.1.4

Multiply

$4^{2}$ by

$1$ .

$e = 12 - \frac{4^{2}}{4 \cdot - 1}$

Step 3.5.2.1.1.5

Factor

$4$ out of

$4^{2}$ .

$e = 12 - \frac{4 \cdot 4}{4 \cdot - 1}$

Step 3.5.2.1.1.6

Move the negative one from the denominator of

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

$e = 12 - (- 1 \cdot 4)$

Step 3.5.2.1.2

Multiply

$- (- 1 \cdot 4)$ .

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Step 3.5.2.1.2.1

Multiply

$- 1$ by

$4$ .

$e = 12 - - 4$

Step 3.5.2.1.2.2

Multiply

$- 1$ by

$- 4$ .

$e = 12 + 4$

Step 3.5.2.2

Add

$12$ and

$4$ .

$e = 16$

Step 3.6

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

$- {(x + 2)}^{2} + 16$ .

$8 \int \frac{1}{\sqrt{- {(x + 2)}^{2} + 16}} d x$

Step 4

Let

$u = x + 2$ . Then

$d u = d x$ . Rewrite using

$u$ and

$d$

$u$ .

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Step 4.1

Let

$u = x + 2$ . Find

$\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate

$x + 2$ .

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

Step 4.1.2

By the Sum Rule, the derivative of

$x + 2$ with respect to

$x$ is

$\frac{d}{d x} [x] + \frac{d}{d x} [2]$ .

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

Step 4.1.3

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 4.1.4

Since

$2$ is constant with respect to

$x$ , the derivative of

$2$ with respect to

$x$ is

$0$ .

$1 + 0$

Step 4.1.5

Add

$1$ and

$0$ .

$1$

Step 4.2

Rewrite the problem using

$u$ and

$d u$ .

$8 \int \frac{1}{\sqrt{- u^{2} + 16}} d u$

Step 5

Simplify the expression.

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Step 5.1

Rewrite

$16$ as

$4^{2}$ .

$8 \int \frac{1}{\sqrt{- u^{2} + 4^{2}}} d u$

Step 5.2

Reorder

$- u^{2}$ and

$4^{2}$ .

$8 \int \frac{1}{\sqrt{4^{2} - u^{2}}} d u$

Step 6

The integral of

$\frac{1}{\sqrt{4^{2} - u^{2}}}$ with respect to

$u$ is

$\arcsin (\frac{u}{4})$

$8 (\arcsin (\frac{u}{4}) + C)$

Step 7

Rewrite

$8 (\arcsin (\frac{u}{4}) + C)$ as

$8 \arcsin (\frac{1}{4} u) + C$ .

$8 \arcsin (\frac{1}{4} u) + C$

Step 8

Replace all occurrences of

$u$ with

$x + 2$ .

$8 \arcsin (\frac{1}{4} (x + 2)) + C$

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

$8 \arcsin (\frac{1}{4} x + \frac{1}{4} \cdot 2) + C$

Step 9.2

Combine

$\frac{1}{4}$ and

$x$ .

$8 \arcsin (\frac{x}{4} + \frac{1}{4} \cdot 2) + C$

Step 9.3

Cancel the common factor of

$2$ .

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Step 9.3.1

Factor

$2$ out of

$4$ .

$8 \arcsin (\frac{x}{4} + \frac{1}{2 (2)} \cdot 2) + C$

Step 9.3.2

Cancel the common factor.

$8 \arcsin (\frac{x}{4} + \frac{1}{2 \cdot 2} \cdot 2) + C$

Step 9.3.3

Rewrite the expression.

$8 \arcsin (\frac{x}{4} + \frac{1}{2}) + C$

Step 10

Reorder terms.

$8 \arcsin (\frac{1}{4} x + \frac{1}{2}) + C$