Calculus Examples

$\int arccsc (x) d x$

Step 1

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

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

Step 2

Simplify.

Tap for more steps...

Step 2.1

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

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

Step 2.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 3

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

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2

Multiply $\int \frac{1}{\sqrt{x^{2} - 1}} d x$ by $1$ .

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

Step 5

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.

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

Step 6

Simplify terms.

Tap for more steps...

Step 6.1

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

Tap for more steps...

Step 6.1.1

Apply pythagorean identity.

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

Step 6.1.2

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

$arccsc (x) x + \int \frac{1}{\tan (t)} (\sec (t) \tan (t)) d t$

Step 6.2

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

Tap for more steps...

Step 6.2.1

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

$arccsc (x) x + \int \frac{1}{\tan (t)} (\tan (t) \sec (t)) d t$

Step 6.2.2

Cancel the common factor.

$arccsc (x) x + \int \frac{1}{\tan (t)} (\tan (t) \sec (t)) d t$

Step 6.2.3

Rewrite the expression.

$arccsc (x) x + \int \sec (t) d t$

Step 7

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

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

Step 8

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

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