Calculus Examples

\int \frac{1}{\sqrt{8 + 2 x - x^{2}}} d x

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

Step 1

Rewrite

8

$8$ as

9

$9$ plus

- 1

$- 1$

\int \frac{1}{\sqrt{9 - 1 + 2 x - x^{2}}} d x

$\int \frac{1}{\sqrt{9 - 1 + 2 x - x^{2}}} d x$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

\int \frac{1}{\sqrt{9 - (1^{2} - 2 x + x^{2})}} d x

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

Step 2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

2 x = 2 \cdot 1 \cdot x

$2 x = 2 \cdot 1 \cdot x$

Step 2.3

Rewrite the polynomial.

\int \frac{1}{\sqrt{9 - (1^{2} - 2 \cdot 1 \cdot x + x^{2})}} d x

$\int \frac{1}{\sqrt{9 - (1^{2} - 2 \cdot 1 \cdot x + x^{2})}} d x$

Step 2.4

Factor using the perfect square trinomial rule

a^{2} - 2 a b + b^{2} = {(a - b)}^{2}

$a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where

a = 1

$a = 1$ and

b = x

$b = x$ .

\int \frac{1}{\sqrt{9 - {(1 - x)}^{2}}} d x

$\int \frac{1}{\sqrt{9 - {(1 - x)}^{2}}} d x$

\int \frac{1}{\sqrt{9 - {(1 - x)}^{2}}} d x

$\int \frac{1}{\sqrt{9 - {(1 - x)}^{2}}} d x$

Step 3

Rewrite

9

$9$ as

3^{2}

$3^{2}$ .

\int \frac{1}{\sqrt{3^{2} - {(1 - x)}^{2}}} d x

$\int \frac{1}{\sqrt{3^{2} - {(1 - x)}^{2}}} d x$

Step 4

Let

u = 1 - x

$u = 1 - x$ . Then

d u = - d x

$d u = - d x$ , so

- d u = d x

$- d u = d x$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

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Step 4.1

Let

u = 1 - x

$u = 1 - x$ . Find

\frac{d u}{d x}

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

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Step 4.1.1

Rewrite.

\frac{- 1}{1}

$\frac{- 1}{1}$

Step 4.1.2

Divide

- 1

$- 1$ by

1

$1$ .

- 1

$- 1$

- 1

$- 1$

Step 4.2

Rewrite the problem using

u

$u$ and

d u

$d u$ .

\int \frac{- 1}{\sqrt{3^{2} - u^{2}}} d u

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

\int \frac{- 1}{\sqrt{3^{2} - u^{2}}} d u

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

Step 5

Since

- 1

$- 1$ is constant with respect to

u

$u$ , move

- 1

$- 1$ out of the integral.

- \int \frac{1}{\sqrt{3^{2} - u^{2}}} d u

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

Step 6

The integral of

\frac{1}{\sqrt{3^{2} - u^{2}}}

$\frac{1}{\sqrt{3^{2} - u^{2}}}$ with respect to

u

$u$ is

\arcsin (\frac{u}{3})

$\arcsin (\frac{u}{3})$

- (\arcsin (\frac{u}{3}) + C)

$- (\arcsin (\frac{u}{3}) + C)$

Step 7

Rewrite

- (\arcsin (\frac{u}{3}) + C)

$- (\arcsin (\frac{u}{3}) + C)$ as

- \arcsin (\frac{1}{3} u) + C

$- \arcsin (\frac{1}{3} u) + C$ .

- \arcsin (\frac{1}{3} u) + C

$- \arcsin (\frac{1}{3} u) + C$

Step 8

Replace all occurrences of

u

$u$ with

1 - x

$1 - x$ .

- \arcsin (\frac{1}{3} (1 - x)) + C

$- \arcsin (\frac{1}{3} (1 - x)) + C$

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

- \arcsin (\frac{1}{3} \cdot 1 + \frac{1}{3} (- x)) + C

$- \arcsin (\frac{1}{3} \cdot 1 + \frac{1}{3} (- x)) + C$

Step 9.2

Multiply

\frac{1}{3}

$\frac{1}{3}$ by

1

$1$ .

- \arcsin (\frac{1}{3} + \frac{1}{3} (- x)) + C

$- \arcsin (\frac{1}{3} + \frac{1}{3} (- x)) + C$

Step 9.3

Combine

\frac{1}{3}

$\frac{1}{3}$ and

x

$x$ .

- \arcsin (\frac{1}{3} - \frac{x}{3}) + C

$- \arcsin (\frac{1}{3} - \frac{x}{3}) + C$

- \arcsin (\frac{1}{3} - \frac{x}{3}) + C

$- \arcsin (\frac{1}{3} - \frac{x}{3}) + C$

Step 10

Reorder terms.

- \arcsin (\frac{1}{3} - \frac{1}{3} x) + C

$- \arcsin (\frac{1}{3} - \frac{1}{3} x) + C$