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Integral of x*arctg(2*x)*dx dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1               
  /               
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 |  x*atan(2*x) dx
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0                 
$$\int\limits_{0}^{1} x \operatorname{atan}{\left(2 x \right)}\, dx$$
Integral(x*atan(2*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. Rewrite the integrand:

  3. 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=4, c=1, context=1/(4*x**2 + 1), symbol=x), True), (ArccothRule(a=1, b=4, c=1, context=1/(4*x**2 + 1), symbol=x), False), (ArctanhRule(a=1, b=4, c=1, context=1/(4*x**2 + 1), symbol=x), False)], context=1/(4*x**2 + 1), symbol=x)

      So, the result is:

    The result is:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                      2          
 |                      x   atan(2*x)   x *atan(2*x)
 | x*atan(2*x) dx = C - - + --------- + ------------
 |                      4       8            2      
/                                                   
$$\int x \operatorname{atan}{\left(2 x \right)}\, dx = C + \frac{x^{2} \operatorname{atan}{\left(2 x \right)}}{2} - \frac{x}{4} + \frac{\operatorname{atan}{\left(2 x \right)}}{8}$$
The graph
The answer [src]
  1   5*atan(2)
- - + ---------
  4       8    
$$- \frac{1}{4} + \frac{5 \operatorname{atan}{\left(2 \right)}}{8}$$
=
=
  1   5*atan(2)
- - + ---------
  4       8    
$$- \frac{1}{4} + \frac{5 \operatorname{atan}{\left(2 \right)}}{8}$$
-1/4 + 5*atan(2)/8
Numerical answer [src]
0.441967948621307
0.441967948621307

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