Calculus Examples

$\int \frac{- 2}{x \sqrt{x^{2} - 36}} d x$

Step 1

Move the negative in front of the fraction.

$\int - \frac{2}{x \sqrt{x^{2} - 36}} d x$

Step 2

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

$- \int \frac{2}{x \sqrt{x^{2} - 36}} d x$

Step 3

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$- (2 \int \frac{1}{x \sqrt{x^{2} - 36}} d x)$

Step 4

Write the expression using exponents.

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Step 4.1

Multiply $2$ by $- 1$ .

$- 2 \int \frac{1}{x \sqrt{x^{2} - 36}} d x$

Step 4.2

Rewrite $36$ as $6^{2}$ .

$- 2 \int \frac{1}{x \sqrt{x^{2} - 6^{2}}} d x$

Step 5

The integral of $\frac{1}{x \sqrt{x^{2} - 6^{2}}}$ with respect to $x$ is $\frac{1}{6} arcsec (| \frac{x}{6} |)$

$- 2 (\frac{1}{6} arcsec (| \frac{x}{6} |) + C)$

Step 6

Simplify the answer.

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Step 6.1

Combine $\frac{1}{6}$ and $arcsec (| \frac{x}{6} |)$ .

$- 2 (\frac{arcsec (| \frac{x}{6} |)}{6} + C)$

Step 6.2

Rewrite $- 2 (\frac{arcsec (| \frac{x}{6} |)}{6} + C)$ as $- 2 \frac{arcsec (\frac{| x |}{6})}{6} + C$ .

$- 2 \frac{arcsec (\frac{| x |}{6})}{6} + C$

Step 6.3

Simplify.

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Step 6.3.1

Combine $- 2$ and $\frac{arcsec (\frac{| x |}{6})}{6}$ .

$\frac{- 2 arcsec (\frac{| x |}{6})}{6} + C$

Step 6.3.2

Cancel the common factor of $- 2$ and $6$ .

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Step 6.3.2.1

Factor $2$ out of $- 2 arcsec (\frac{| x |}{6})$ .

$\frac{2 (- arcsec (\frac{| x |}{6}))}{6} + C$

Step 6.3.2.2

Cancel the common factors.

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Step 6.3.2.2.1

Factor $2$ out of $6$ .

$\frac{2 (- arcsec (\frac{| x |}{6}))}{2 (3)} + C$

Step 6.3.2.2.2

Cancel the common factor.

$\frac{2 (- arcsec (\frac{| x |}{6}))}{2 \cdot 3} + C$

Step 6.3.2.2.3

Rewrite the expression.

$\frac{- arcsec (\frac{| x |}{6})}{3} + C$

Step 6.3.3

Move the negative in front of the fraction.

$- \frac{arcsec (\frac{| x |}{6})}{3} + C$

Step 6.4

Reorder terms.

$- \frac{1}{3} arcsec (\frac{1}{6} | x |) + C$