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Solving integrals/ x*arcsin(1/x)dx

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

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The solution

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∫01xasin⁡(1x) dx\int\limits_{0}^{1} x \operatorname{asin}{\left(\frac{1}{x} \right)}\, dx 
Integral(x*asin(1/x), (x, 0, 1))
Detail solution

  1. Use integration by parts:

    ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

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

    Then du⁡(x)=−1x21−1x2\operatorname{du}{\left(x \right)} = - \frac{1}{x^{2} \sqrt{1 - \frac{1}{x^{2}}}}.

    To find v(x)v{\left(x \right)}:

    1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

      ∫x dx=x22\int x\, dx = \frac{x^{2}}{2}

    Now evaluate the sub-integral.

  2. The integral of a constant times a function is the constant times the integral of the function:

    ∫(−121−1x2) dx=−∫11−1x2 dx2\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

      {x2−1for ∣x2∣>1i1−x2otherwise\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: −{x2−1for ∣x2∣>1i1−x2otherwise2- \frac{\begin{cases} \sqrt{x^{2} - 1} & \text{for}\: \left|{x^{2}}\right| > 1 \\i \sqrt{1 - x^{2}} & \text{otherwise} \end{cases}}{2}

  3. Now simplify:

    {x2asin⁡(1x)+x2−12for ∣x2∣>1x2asin⁡(1x)+i1−x22otherwise\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}

  4. Add the constant of integration:

    {x2asin⁡(1x)+x2−12for ∣x2∣>1x2asin⁡(1x)+i1−x22otherwise+constant\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:

{x2asin⁡(1x)+x2−12for ∣x2∣>1x2asin⁡(1x)+i1−x22otherwise+constant\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\
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 |       /1\          \I*\/  1 - x     otherwise            \x/
 | x*asin|-| dx = C + ---------------------------- + ----------
 |       \x/                       2                     2     
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/
∫xasin⁡(1x) dx=C+x2asin⁡(1x)2+{x2−1for ∣x2∣>1i1−x2otherwise2\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
1.0000000.9997500.9997750.9998000.9998250.9998500.9998750.9999000.9999250.9999500.9999750.02.0
Plot the graph f(x)
The answer [src] 
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∫01xasin⁡(1x) dx\int\limits_{0}^{1} x \operatorname{asin}{\left(\frac{1}{x} \right)}\, dx 
=
= 
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∫01xasin⁡(1x) dx\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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