Calculus Examples

$\int arcsec (x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = arcsec (x)$ and $d v = 1$ .

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

Step 2

Simplify.

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Step 2.1

Combine $x$ and $\frac{1}{x \sqrt{x^{2} - 1}}$ .

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

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 3

Let $x = \sec (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \sec (t) \tan (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\sec (t) \tan (t)$ is positive.

$arcsec (x) x - \int \frac{1}{\sqrt{\sec^{2} (t) - 1}} (\sec (t) \tan (t)) d t$

Step 4

Simplify terms.

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Step 4.1

Simplify $\sqrt{\sec^{2} (t) - 1}$ .

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Step 4.1.1

Apply pythagorean identity.

$arcsec (x) x - \int \frac{1}{\sqrt{\tan^{2} (t)}} (\sec (t) \tan (t)) d t$

Step 4.1.2

Pull terms out from under the radical, assuming positive real numbers.

$arcsec (x) x - \int \frac{1}{\tan (t)} (\sec (t) \tan (t)) d t$

Step 4.2

Cancel the common factor of $\tan (t)$ .

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Step 4.2.1

Factor $\tan (t)$ out of $\sec (t) \tan (t)$ .

$arcsec (x) x - \int \frac{1}{\tan (t)} (\tan (t) \sec (t)) d t$

Step 4.2.2

Cancel the common factor.

$arcsec (x) x - \int \frac{1}{\tan (t)} (\tan (t) \sec (t)) d t$

Step 4.2.3

Rewrite the expression.

$arcsec (x) x - \int \sec (t) d t$

Step 5

The integral of $\sec (t)$ with respect to $t$ is $\ln (| \sec (t) + \tan (t) |)$ .

$arcsec (x) x - (\ln (| \sec (t) + \tan (t) |) + C)$

Step 6

Simplify.

$arcsec (x) x - \ln (| \sec (t) + \tan (t) |) + C$

Step 7

Replace all occurrences of $t$ with $arcsec (x)$ .

$arcsec (x) x - \ln (| \sec (arcsec (x)) + \tan (arcsec (x)) |) + C$