Mister Exam

Integral of arctg(2x) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

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


The answer is:

The answer (Indefinite) [src]
  /                      /       2\              
 |                    log\1 + 4*x /              
 | atan(2*x) dx = C - ------------- + x*atan(2*x)
 |                          4                    
/                                                
$${{2\,x\,\arctan \left(2\,x\right)-{{\log \left(4\,x^2+1\right) }\over{2}}}\over{2}}$$
The graph
The answer [src]
  log(5)          
- ------ + atan(2)
    4             
$$-{{\log 5-4\,\arctan 2}\over{4}}$$
=
=
  log(5)          
- ------ + atan(2)
    4             
$$- \frac{\log{\left(5 \right)}}{4} + \operatorname{atan}{\left(2 \right)}$$
Numerical answer [src]
0.704789239685565
0.704789239685565
The graph
Integral of arctg(2x) dx

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