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Identical expressions
x*arcsin(one /x)dx
x multiply by arc sinus of (1 divide by x)dx
x multiply by arc sinus of (one divide by x)dx
xarcsin(1/x)dx
xarcsin1/xdx
x*arcsin(1 divide by x)dx

Integral of x*arcsin(1/x)dx dx

The solution

You have entered [src]

  1             
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 |  x*asin|-| dx
 |        \x/   
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0

$$\int\limits_{0}^{1} x \operatorname{asin}{\left(\frac{1}{x} \right)}\, dx$$

Integral(x*asin(1/x), (x, 0, 1))

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 $\operatorname{dv}{\left(x \right)} = x$ .

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 $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x\, dx = \frac{x^{2}}{2}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

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

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

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

The answer is:

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

The answer (Indefinite) [src]

                      /   _________                            
                      |  /       2        | 2|                 
                      |\/  -1 + x     for |x | > 1             
                      <                                        
  /                   |     ________                  2     /1\
 |                    |    /      2                  x *asin|-|
 |       /1\          \I*\/  1 - x     otherwise            \x/
 | x*asin|-| dx = C + ---------------------------- + ----------
 |       \x/                       2                     2     
 |                                                             
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

  1             
  /             
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 |        /1\   
 |  x*asin|-| dx
 |        \x/   
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/               
0

$$\int\limits_{0}^{1} x \operatorname{asin}{\left(\frac{1}{x} \right)}\, dx$$

  1             
  /             
 |              
 |        /1\   
 |  x*asin|-| dx
 |        \x/   
 |              
/               
0

$$\int\limits_{0}^{1} x \operatorname{asin}{\left(\frac{1}{x} \right)}\, dx$$

Integral(x*asin(1/x), (x, 0, 1))

Numerical answer [src]

(0.785398163397448 - 0.5j)

(0.785398163397448 - 0.5j)

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

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How to use it?

Identical expressions

Integral of x*arcsin(1/x)dx dx

Limits of integration:

The graph:

Piecewise:

The solution