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x*arctg(x)

Integral of x*arctg(x) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1             
  /             
 |              
 |  x*atan(x) dx
 |              
/               
0               
$$\int\limits_{0}^{1} x \operatorname{atan}{\left(x \right)}\, dx$$
Integral(x*atan(x), (x, 0, 1))
Detail solution
  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:

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. The integral of a constant is the constant times the variable of integration:

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

          PiecewiseRule(subfunctions=[(ArctanRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), True), (ArccothRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), False), (ArctanhRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), False)], context=1/(x**2 + 1), symbol=x)

        So, the result is:

      The result is:

    So, the result is:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                  2        
 |                    atan(x)   x   x *atan(x)
 | x*atan(x) dx = C + ------- - - + ----------
 |                       2      2       2     
/                                             
$$\int x \operatorname{atan}{\left(x \right)}\, dx = C + \frac{x^{2} \operatorname{atan}{\left(x \right)}}{2} - \frac{x}{2} + \frac{\operatorname{atan}{\left(x \right)}}{2}$$
The graph
The answer [src]
  1   pi
- - + --
  2   4 
$$- \frac{1}{2} + \frac{\pi}{4}$$
=
=
  1   pi
- - + --
  2   4 
$$- \frac{1}{2} + \frac{\pi}{4}$$
-1/2 + pi/4
Numerical answer [src]
0.285398163397448
0.285398163397448
The graph
Integral of x*arctg(x) dx

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