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1/sqrt(x^2+4)

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                 
  /                 
 |                  
 |         1        
 |  1*----------- dx
 |       ________   
 |      /  2        
 |    \/  x  + 4    
 |                  
/                   
0                   
$$\int\limits_{0}^{1} 1 \cdot \frac{1}{\sqrt{x^{2} + 4}}\, dx$$
Integral(1/sqrt(x^2 + 4), (x, 0, 1))
Detail solution

    TrigSubstitutionRule(theta=_theta, func=2*tan(_theta), rewritten=sec(_theta), substep=RewriteRule(rewritten=(tan(_theta)*sec(_theta) + sec(_theta)**2)/(tan(_theta) + sec(_theta)), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=tan(_theta) + sec(_theta), constant=1, substep=ReciprocalRule(func=_u, context=1/_u, symbol=_u), context=(tan(_theta)*sec(_theta) + sec(_theta)**2)/(tan(_theta) + sec(_theta)), symbol=_theta)], context=(tan(_theta)*sec(_theta) + sec(_theta)**2)/(tan(_theta) + sec(_theta)), symbol=_theta), context=sec(_theta), symbol=_theta), restriction=True, context=1/sqrt(x**2 + 4), symbol=x)

  1. Now simplify:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                          /     ________    \
 |                           |    /      2     |
 |        1                  |   /      x     x|
 | 1*----------- dx = C + log|  /   1 + --  + -|
 |      ________             \\/        4     2/
 |     /  2                                     
 |   \/  x  + 4                                 
 |                                              
/                                               
$${\rm asinh}\; \left({{x}\over{2}}\right)$$
The graph
The answer [src]
asinh(1/2)
$${\rm asinh}\; \left({{1}\over{2}}\right)$$
=
=
asinh(1/2)
$$\operatorname{asinh}{\left(\frac{1}{2} \right)}$$
Numerical answer [src]
0.481211825059603
0.481211825059603
The graph
Integral of 1/sqrt(x^2+4) dx

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