Integral of xarctan(2x) dx

The solution

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  1               
  /               
 |                
 |  x*atan(2*x) dx
 |                
/                 
0

$$\int\limits_{0}^{1} x \operatorname{atan}{\left(2 x \right)}\, dx$$

Integral(x*atan(2*x), (x, 0, 1))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = \operatorname{atan}{\left(2 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = x$ .

Then $\operatorname{du}{\left(x \right)} = \frac{2}{4 x^{2} + 1}$ .

To find $v{\left(x \right)}$ :
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x\, dx = \frac{x^{2}}{2}$
Now evaluate the sub-integral.
Rewrite the integrand:

$\frac{x^{2}}{4 x^{2} + 1} = \frac{1}{4} - \frac{1}{4 \cdot \left(4 x^{2} + 1\right)}$
Integrate term-by-term:
1. The integral of a constant is the constant times the variable of integration:
  
  $\int \frac{1}{4}\, dx = \frac{x}{4}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{4 \cdot \left(4 x^{2} + 1\right)}\right)\, dx = - \frac{\int \frac{1}{4 x^{2} + 1}\, dx}{4}$
  1. The integral of $\frac{1}{x^{2} + 1}$ is $\frac{\operatorname{atan}{\left(2 x \right)}}{2}$ .
  So, the result is: $- \frac{\operatorname{atan}{\left(2 x \right)}}{8}$
The result is: $\frac{x}{4} - \frac{\operatorname{atan}{\left(2 x \right)}}{8}$
Add the constant of integration:

$\frac{x^{2} \operatorname{atan}{\left(2 x \right)}}{2} - \frac{x}{4} + \frac{\operatorname{atan}{\left(2 x \right)}}{8}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \operatorname{atan}{\left(2 x \right)}}{2} - \frac{x}{4} + \frac{\operatorname{atan}{\left(2 x \right)}}{8}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                      2          
 |                      x   atan(2*x)   x *atan(2*x)
 | x*atan(2*x) dx = C - - + --------- + ------------
 |                      4       8            2      
/

$${{x^2\,\arctan \left(2\,x\right)}\over{2}}+{{\arctan \left(2\,x \right)}\over{8}}-{{x}\over{4}}$$

Simplify

The answer [src]

  1   5*atan(2)
- - + ---------
  4       8

$${{5\,\arctan 2-2}\over{8}}$$

  1   5*atan(2)
- - + ---------
  4       8

$$- \frac{1}{4} + \frac{5 \operatorname{atan}{\left(2 \right)}}{8}$$

Упростить

Numerical answer [src]

0.441967948621307

0.441967948621307

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

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Identical expressions

Integral of xarctan(2x) dx

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