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Integral of arctg(x/2)dx dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  2           
  /           
 |            
 |      /x\   
 |  atan|-| dx
 |      \2/   
 |            
/             
0             
$$\int\limits_{0}^{2} \operatorname{atan}{\left(\frac{x}{2} \right)}\, dx$$
Integral(atan(x/2), (x, 0, 2))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let .

      Then let and substitute :

      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 a constant is the constant times the variable of integration:

          Now evaluate the sub-integral.

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

          1. Let .

            Then let and substitute :

            1. The integral of is .

            Now substitute back in:

          So, the result is:

        So, the result is:

      Now substitute back in:

    Method #2

    1. Use integration by parts:

      Let and let .

      Then .

      To find :

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

      Now evaluate the sub-integral.

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

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

        1. Let .

          Then let and substitute :

          1. The integral of is .

          Now substitute back in:

        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\            
 |     /x\             |    x |         /x\
 | atan|-| dx = C - log|1 + --| + x*atan|-|
 |     \2/             \    4 /         \2/
 |                                         
/                                          
$$\int \operatorname{atan}{\left(\frac{x}{2} \right)}\, dx = C + x \operatorname{atan}{\left(\frac{x}{2} \right)} - \log{\left(\frac{x^{2}}{4} + 1 \right)}$$
The graph
The answer [src]
pi                  
-- - log(8) + log(4)
2                   
$$- \log{\left(8 \right)} + \log{\left(4 \right)} + \frac{\pi}{2}$$
=
=
pi                  
-- - log(8) + log(4)
2                   
$$- \log{\left(8 \right)} + \log{\left(4 \right)} + \frac{\pi}{2}$$
pi/2 - log(8) + log(4)
Numerical answer [src]
0.877649146234951
0.877649146234951

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