Calculus Examples

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

Step 1

Rewrite $8$ as $9$ plus $- 1$

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

Step 2

Factor using the perfect square rule.

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Step 2.1

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

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

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.

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

Step 2.3

Rewrite the polynomial.

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

Step 2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 1$ and $b = x$ .

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

Step 3

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

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

Step 4

Let $u = 1 - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

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

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Step 4.1.1

Rewrite.

$\frac{- 1}{1}$

Step 4.1.2

Divide $- 1$ by $1$ .

$- 1$

Step 4.2

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

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

Step 5

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

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

Step 6

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

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

Step 7

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

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

Step 8

Replace all occurrences of $u$ with $1 - x$ .

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

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Multiply $\frac{1}{3}$ by $1$ .

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

Step 9.3

Combine $\frac{1}{3}$ and $x$ .

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

Step 10

Reorder terms.

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