Mister Exam

Other calculators

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

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1            
  /            
 |             
 |     x       
 |  -------- dx
 |     2       
 |  4*x  + 1   
 |             
/              
0
$$\int\limits_{0}^{1} \frac{x}{4 x^{2} + 1}\, dx$$ 
Integral(x/(4*x^2 + 1), (x, 0, 1))
Detail solution
We have the integral:
  /           
 |            
 |    x       
 | -------- dx
 |    2       
 | 4*x  + 1   
 |            
/
Rewrite the integrand
           /    4*2*x     \              
           |--------------|       /0\    
           |   2          |       |-|    
   x       \4*x  + 0*x + 1/       \1/    
-------- = ---------------- + -----------
   2              8                 2    
4*x  + 1                      (-2*x)  + 1
or
  /             
 |              
 |    x         
 | -------- dx  
 |    2        =
 | 4*x  + 1     
 |              
/               
  
  /                 
 |                  
 |     4*2*x        
 | -------------- dx
 |    2             
 | 4*x  + 0*x + 1   
 |                  
/                   
--------------------
         8
In the integral
  /                 
 |                  
 |     4*2*x        
 | -------------- dx
 |    2             
 | 4*x  + 0*x + 1   
 |                  
/                   
--------------------
         8
do replacement
       2
u = 4*x
then
the integral =
  /                     
 |                      
 |   1                  
 | ----- du             
 | 1 + u                
 |                      
/             log(1 + u)
----------- = ----------
     8            8
do backward replacement
  /                                 
 |                                  
 |     4*2*x                        
 | -------------- dx                
 |    2                             
 | 4*x  + 0*x + 1                   
 |                        /       2\
/                      log\1 + 4*x /
-------------------- = -------------
         8                   8
In the integral
0
do replacement
v = -2*x
then
the integral =
True
do backward replacement
True
Solution is:
       /       2\
    log\1 + 4*x /
C + -------------
          8
The answer (Indefinite) [src] 
  /                               
 |                      /       2\
 |    x              log\1 + 4*x /
 | -------- dx = C + -------------
 |    2                    8      
 | 4*x  + 1                       
 |                                
/
$$\int \frac{x}{4 x^{2} + 1}\, dx = C + \frac{\log{\left(4 x^{2} + 1 \right)}}{8}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
log(5)
------
  8
$$\frac{\log{\left(5 \right)}}{8}$$ 
=
= 
log(5)
------
  8
$$\frac{\log{\left(5 \right)}}{8}$$ 
log(5)/8
Numerical answer [src] 
0.201179739054263
0.201179739054263
The graph
Integral of x/(4x^2+1) dx

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

    © Mister Exam – Calculator