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Integral of x*asin(x)/3 dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1             
  /             
 |              
 |  x*asin(x)   
 |  --------- dx
 |      3       
 |              
/               
0               
$$\int\limits_{0}^{1} \frac{x \operatorname{asin}{\left(x \right)}}{3}\, dx$$
Integral((x*asin(x))/3, (x, 0, 1))
Detail solution
  1. The integral of a constant times a function is the constant times the integral of the function:

    1. Use integration by parts:

      Let and let .

      Then .

      To find :

      1. The integral of is when :

      Now evaluate the sub-integral.

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

        TrigSubstitutionRule(theta=_theta, func=sin(_theta), rewritten=sin(_theta)**2, substep=RewriteRule(rewritten=1/2 - cos(2*_theta)/2, substep=AddRule(substeps=[ConstantRule(constant=1/2, context=1/2, symbol=_theta), ConstantTimesRule(constant=-1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=-cos(2*_theta)/2, symbol=_theta)], context=1/2 - cos(2*_theta)/2, symbol=_theta), context=sin(_theta)**2, symbol=_theta), restriction=(x > -1) & (x < 1), context=x**2/sqrt(1 - x**2), symbol=x)

      So, the result is:

    So, the result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
                      /               ________                                     
                      |              /      2                                      
  /                    -1, x < 1)    2        
 | x*asin(x)          \   2            2                                 x *asin(x)
 | --------- dx = C - ------------------------------------------------ + ----------
 |     3                                     6                               6     
 |                                                                                 
/                                                                                  
$$\int \frac{x \operatorname{asin}{\left(x \right)}}{3}\, dx = C + \frac{x^{2} \operatorname{asin}{\left(x \right)}}{6} - \frac{\begin{cases} - \frac{x \sqrt{1 - x^{2}}}{2} + \frac{\operatorname{asin}{\left(x \right)}}{2} & \text{for}\: x > -1 \wedge x < 1 \end{cases}}{6}$$
The graph
The answer [src]
pi
--
24
$$\frac{\pi}{24}$$
=
=
pi
--
24
$$\frac{\pi}{24}$$
pi/24
Numerical answer [src]
0.130899693899575
0.130899693899575

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