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1/(25+4x^2)
Solving integrals/ 1/(25+4x^2)

Integral of 1/(25+4x^2) dx

Limits of integration:

from to
v

The graph:

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Piecewise:

The solution

You have entered [src] 
  1             
  /             
 |              
 |      1       
 |  --------- dx
 |          2   
 |  25 + 4*x    
 |              
/               
0
$$\int\limits_{0}^{1} \frac{1}{4 x^{2} + 25}\, dx$$ 
Integral(1/(25 + 4*x^2), (x, 0, 1))
Detail solution
We have the integral:
  /            
 |             
 |     1       
 | --------- dx
 |         2   
 | 25 + 4*x    
 |             
/
Rewrite the integrand
    1              1        
--------- = ----------------
        2      /      2    \
25 + 4*x       |/-2*x\     |
            25*||----|  + 1|
               \\ 5  /     /
or
  /              
 |               
 |     1         
 | --------- dx  
 |         2    =
 | 25 + 4*x      
 |               
/                
  
  /              
 |               
 |      1        
 | ----------- dx
 |       2       
 | /-2*x\        
 | |----|  + 1   
 | \ 5  /        
 |               
/                
-----------------
        25
In the integral
  /              
 |               
 |      1        
 | ----------- dx
 |       2       
 | /-2*x\        
 | |----|  + 1   
 | \ 5  /        
 |               
/                
-----------------
        25
do replacement
    -2*x
v = ----
     5
then
the integral =
  /                   
 |                    
 |   1                
 | ------ dv          
 |      2             
 | 1 + v              
 |                    
/              atan(v)
------------ = -------
     25           25
do backward replacement
  /                          
 |                           
 |      1                    
 | ----------- dx            
 |       2                   
 | /-2*x\                    
 | |----|  + 1               
 | \ 5  /               /2*x\
 |                  atan|---|
/                       \ 5 /
----------------- = ---------
        25              10
Solution is:
        /2*x\
    atan|---|
        \ 5 /
C + ---------
        10
The answer (Indefinite) [src] 
  /                       /2*x\
 |                    atan|---|
 |     1                  \ 5 /
 | --------- dx = C + ---------
 |         2              10   
 | 25 + 4*x                    
 |                             
/
$$\int \frac{1}{4 x^{2} + 25}\, dx = C + \frac{\operatorname{atan}{\left(\frac{2 x}{5} \right)}}{10}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
atan(2/5)
---------
    10
$$\frac{\operatorname{atan}{\left(\frac{2}{5} \right)}}{10}$$ 
=
= 
atan(2/5)
---------
    10
$$\frac{\operatorname{atan}{\left(\frac{2}{5} \right)}}{10}$$ 
atan(2/5)/10
Numerical answer [src] 
0.0380506377112365
0.0380506377112365
The graph
Integral of 1/(25+4x^2) dx

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

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