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Identical expressions
(atan two x)/ one +4x^2
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atan2x/1+4x2
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atan2x/1+4x^2
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(atan2x)/1+4x^2dx
Similar expressions
(atan2x)/1-4x^2

Integral of (atan2x)/1+4x^2 dx

The solution

You have entered [src]

  1                      
  /                      
 |                       
 |  /atan(2*x)      2\   
 |  |--------- + 4*x | dx
 |  \    1           /   
 |                       
/                        
0

$$\int\limits_{0}^{1} \left(4 x^{2} + \frac{\operatorname{atan}{\left(2 x \right)}}{1}\right)\, dx$$

Integral(atan(2*x)/1 + 4*x^2, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 4 x^{2}\, dx = 4 \int x^{2}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{2}\, dx = \frac{x^{3}}{3}$
  So, the result is: $\frac{4 x^{3}}{3}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{\operatorname{atan}{\left(2 x \right)}}{1}\, dx = \int \operatorname{atan}{\left(2 x \right)}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $x \operatorname{atan}{\left(2 x \right)} - \frac{\log{\left(4 x^{2} + 1 \right)}}{4}$
  So, the result is: $x \operatorname{atan}{\left(2 x \right)} - \frac{\log{\left(4 x^{2} + 1 \right)}}{4}$
The result is: $\frac{4 x^{3}}{3} + x \operatorname{atan}{\left(2 x \right)} - \frac{\log{\left(4 x^{2} + 1 \right)}}{4}$
Add the constant of integration:

$\frac{4 x^{3}}{3} + x \operatorname{atan}{\left(2 x \right)} - \frac{\log{\left(4 x^{2} + 1 \right)}}{4}+ \mathrm{constant}$

The answer is:

$\frac{4 x^{3}}{3} + x \operatorname{atan}{\left(2 x \right)} - \frac{\log{\left(4 x^{2} + 1 \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                              
 |                                /       2\      3              
 | /atan(2*x)      2\          log\1 + 4*x /   4*x               
 | |--------- + 4*x | dx = C - ------------- + ---- + x*atan(2*x)
 | \    1           /                4          3                
 |                                                               
/

$$\int \left(4 x^{2} + \frac{\operatorname{atan}{\left(2 x \right)}}{1}\right)\, dx = C + \frac{4 x^{3}}{3} + x \operatorname{atan}{\left(2 x \right)} - \frac{\log{\left(4 x^{2} + 1 \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

4   log(5)          
- - ------ + atan(2)
3     4

$$- \frac{\log{\left(5 \right)}}{4} + \operatorname{atan}{\left(2 \right)} + \frac{4}{3}$$

4   log(5)          
- - ------ + atan(2)
3     4

$$- \frac{\log{\left(5 \right)}}{4} + \operatorname{atan}{\left(2 \right)} + \frac{4}{3}$$

4/3 - log(5)/4 + atan(2)

Numerical answer [src]

2.0381225730189

2.0381225730189

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

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Integral of (atan2x)/1+4x^2 dx

Limits of integration:

The graph:

Piecewise:

The solution