Mister Exam

Integral of xarctg4x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1               
  /               
 |                
 |  x*atan(4*x) dx
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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:

          PiecewiseRule(subfunctions=[(ArctanRule(a=1, b=16, c=1, context=1/(16*x**2 + 1), symbol=x), True), (ArccothRule(a=1, b=16, c=1, context=1/(16*x**2 + 1), symbol=x), False), (ArctanhRule(a=1, b=16, c=1, context=1/(16*x**2 + 1), symbol=x), False)], context=1/(16*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          
 |                      x   atan(4*x)   x *atan(4*x)
 | x*atan(4*x) dx = C - - + --------- + ------------
 |                      8       32           2      
/                                                   
$$\int x \operatorname{atan}{\left(4 x \right)}\, dx = C + \frac{x^{2} \operatorname{atan}{\left(4 x \right)}}{2} - \frac{x}{8} + \frac{\operatorname{atan}{\left(4 x \right)}}{32}$$
The graph
The answer [src]
  1   17*atan(4)
- - + ----------
  8       32    
$$- \frac{1}{8} + \frac{17 \operatorname{atan}{\left(4 \right)}}{32}$$
=
=
  1   17*atan(4)
- - + ----------
  8       32    
$$- \frac{1}{8} + \frac{17 \operatorname{atan}{\left(4 \right)}}{32}$$
-1/8 + 17*atan(4)/32
Numerical answer [src]
0.579340633823642
0.579340633823642
The graph
Integral of xarctg4x dx

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