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x*arctg(4x)

Integral of x*arctg(4x) dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src]
  1               
  /               
 |                
 |  x*atan(4*x) dx
 |                
/                 
0                 
$$\int\limits_{0}^{1} x \operatorname{atan}{\left(4 x \right)}\, dx$$
Integral(x*atan(4*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:

        1. The integral of is .

        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          
 |                      x   atan(4*x)   x *atan(4*x)
 | x*atan(4*x) dx = C - - + --------- + ------------
 |                      8       32           2      
/                                                   
$${{x^2\,\arctan \left(4\,x\right)}\over{2}}-2\,\left({{x}\over{16}}- {{\arctan \left(4\,x\right)}\over{64}}\right)$$
The graph
The answer [src]
  1   17*atan(4)
- - + ----------
  8       32    
$${{17\,\arctan 4-4}\over{32}}$$
=
=
  1   17*atan(4)
- - + ----------
  8       32    
$$- \frac{1}{8} + \frac{17 \operatorname{atan}{\left(4 \right)}}{32}$$
Numerical answer [src]
0.579340633823642
0.579340633823642
The graph
Integral of x*arctg(4x) dx

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