Calculus Examples

$\int_{5}^{8} \frac{1}{x \sqrt{x^{2} - 16}} d x$

Step 1

Rewrite $16$ as $4^{2}$ .

$\int_{5}^{8} \frac{1}{x \sqrt{x^{2} - 4^{2}}} d x$

Step 2

The integral of $\frac{1}{x \sqrt{x^{2} - 4^{2}}}$ with respect to $x$ is $\frac{1}{4} arcsec (| \frac{x}{4} |)$

$\frac{1}{4} arcsec (| \frac{x}{4} |)]_{5}^{8}$

Step 3

Simplify the answer.

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Step 3.1

Combine $\frac{1}{4}$ and $arcsec (| \frac{x}{4} |)$ .

$\frac{arcsec (| \frac{x}{4} |)}{4}]_{5}^{8}$

Step 3.2

Substitute and simplify.

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Step 3.2.1

Evaluate $\frac{arcsec (| \frac{x}{4} |)}{4}$ at $8$ and at $5$ .

$(\frac{arcsec (| \frac{8}{4} |)}{4}) - \frac{arcsec (| \frac{5}{4} |)}{4}$

Step 3.2.2

Cancel the common factor of $8$ and $4$ .

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Step 3.2.2.1

Factor $4$ out of $8$ .

$\frac{arcsec (| \frac{4 \cdot 2}{4} |)}{4} - \frac{arcsec (| \frac{5}{4} |)}{4}$

Step 3.2.2.2

Cancel the common factors.

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Step 3.2.2.2.1

Factor $4$ out of $4$ .

$\frac{arcsec (| \frac{4 \cdot 2}{4 (1)} |)}{4} - \frac{arcsec (| \frac{5}{4} |)}{4}$

Step 3.2.2.2.2

Cancel the common factor.

$\frac{arcsec (| \frac{4 \cdot 2}{4 \cdot 1} |)}{4} - \frac{arcsec (| \frac{5}{4} |)}{4}$

Step 3.2.2.2.3

Rewrite the expression.

$\frac{arcsec (| \frac{2}{1} |)}{4} - \frac{arcsec (| \frac{5}{4} |)}{4}$

Step 3.2.2.2.4

Divide $2$ by $1$ .

$\frac{arcsec (| 2 |)}{4} - \frac{arcsec (| \frac{5}{4} |)}{4}$

Step 3.3

Simplify.

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Step 3.3.1

Combine the numerators over the common denominator.

$\frac{arcsec (| 2 |) - arcsec (| \frac{5}{4} |)}{4}$

Step 3.3.2

Simplify each term.

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Step 3.3.2.1

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{arcsec (2) - arcsec (| \frac{5}{4} |)}{4}$

Step 3.3.2.2

The exact value of $arcsec (2)$ is $\frac{π}{3}$ .

$\frac{\frac{π}{3} - arcsec (| \frac{5}{4} |)}{4}$

Step 3.3.2.3

$\frac{5}{4}$ is approximately $1.25$ which is positive so remove the absolute value

$\frac{\frac{π}{3} - arcsec (\frac{5}{4})}{4}$

Step 3.3.2.4

Evaluate $arcsec (\frac{5}{4})$ .

$\frac{\frac{π}{3} - 1 \cdot 0.6435011}{4}$

Step 3.3.2.5

Multiply $- 1$ by $0.6435011$ .

$\frac{\frac{π}{3} - 0.6435011}{4}$

Step 3.3.3

Subtract $0.6435011$ from $\frac{π}{3}$ .

$\frac{0.40369644}{4}$

Step 3.3.4

Divide $0.40369644$ by $4$ .

$0.10092411$

Step 4