Mister Exam

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How to use it?
Integral of d{x}:
Integral of x^(1/3)
Integral of sin(x)^4
Integral of cos(x)^3
Integral of (e^(x^2))
Graphing y =:
arcsin(1/x)
Derivative of:
arcsin(1/x)
Identical expressions
arcsin(one /x)
arc sinus of (1 divide by x)
arc sinus of (one divide by x)
arcsin1/x
arcsin(1 divide by x)
arcsin(1/x)dx

Integral of arcsin(1/x) dx

The solution

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 oo           
  /           
 |            
 |      /1\   
 |  asin|-| dx
 |      \x/   
 |            
/             
2

$$\int\limits_{2}^{\infty} \operatorname{asin}{\left(\frac{1}{x} \right)}\, dx$$

Integral(asin(1/x), (x, 2, oo))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = \operatorname{asin}{\left(\frac{1}{x} \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$ .

Then $\operatorname{du}{\left(x \right)} = - \frac{1}{x^{2} \sqrt{1 - \frac{1}{x^{2}}}}$ .

To find $v{\left(x \right)}$ :
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{1}{x \sqrt{1 - \frac{1}{x^{2}}}}\right)\, dx = - \int \frac{1}{x \sqrt{1 - \frac{1}{x^{2}}}}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\begin{cases} \operatorname{acosh}{\left(x \right)} & \text{for}\: \left|{x^{2}}\right| > 1 \\- i \operatorname{asin}{\left(x \right)} & \text{otherwise} \end{cases}$
So, the result is: $- \begin{cases} \operatorname{acosh}{\left(x \right)} & \text{for}\: \left|{x^{2}}\right| > 1 \\- i \operatorname{asin}{\left(x \right)} & \text{otherwise} \end{cases}$
Now simplify:

$\begin{cases} x \operatorname{asin}{\left(\frac{1}{x} \right)} + \operatorname{acosh}{\left(x \right)} & \text{for}\: \left|{x^{2}}\right| > 1 \\x \operatorname{asin}{\left(\frac{1}{x} \right)} - i \operatorname{asin}{\left(x \right)} & \text{otherwise} \end{cases}$
Add the constant of integration:

$\begin{cases} x \operatorname{asin}{\left(\frac{1}{x} \right)} + \operatorname{acosh}{\left(x \right)} & \text{for}\: \left|{x^{2}}\right| > 1 \\x \operatorname{asin}{\left(\frac{1}{x} \right)} - i \operatorname{asin}{\left(x \right)} & \text{otherwise} \end{cases}+ \mathrm{constant}$

The answer is:

$\begin{cases} x \operatorname{asin}{\left(\frac{1}{x} \right)} + \operatorname{acosh}{\left(x \right)} & \text{for}\: \left|{x^{2}}\right| > 1 \\x \operatorname{asin}{\left(\frac{1}{x} \right)} - i \operatorname{asin}{\left(x \right)} & \text{otherwise} \end{cases}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                        
 |                              //                | 2|    \
 |     /1\                /1\   || acosh(x)   for |x | > 1|
 | asin|-| dx = C + x*asin|-| + |<                        |
 |     \x/                \x/   ||-I*asin(x)   otherwise  |
 |                              \\                        /
/

$$\int \operatorname{asin}{\left(\frac{1}{x} \right)}\, dx = C + x \operatorname{asin}{\left(\frac{1}{x} \right)} + \begin{cases} \operatorname{acosh}{\left(x \right)} & \text{for}\: \left|{x^{2}}\right| > 1 \\- i \operatorname{asin}{\left(x \right)} & \text{otherwise} \end{cases}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo - acosh(2)

$$- \operatorname{acosh}{\left(2 \right)} + \infty$$

oo - acosh(2)

$$- \operatorname{acosh}{\left(2 \right)} + \infty$$

oo - acosh(2)

Use the examples entering the upper and lower limits of integration.

Other calculators

How to use it?

Identical expressions

Integral of arcsin(1/x) dx

Limits of integration:

The graph:

Piecewise:

The solution