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dx/Sqrt[25x^2+4]
Solving integrals/ dx/Sqrt[25x^2+4]

Integral of dx/Sqrt[25x^2+4] dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
  1                    
  /                    
 |                     
 |          1          
 |  1*-------------- dx
 |       ___________   
 |      /     2        
 |    \/  25*x  + 4    
 |                     
/                      
0
∫011⋅125x2+4 dx\int\limits_{0}^{1} 1 \cdot \frac{1}{\sqrt{25 x^{2} + 4}}\, dx
Simplify
Detail solution

    TrigSubstitutionRule(theta=_theta, func=2*tan(_theta)/5, rewritten=sec(_theta)/5, substep=ConstantTimesRule(constant=1/5, other=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), context=sec(_theta)/5, symbol=_theta), restriction=True, context=1/sqrt(25*x**2 + 4), symbol=x)

  1. Now simplify:

    log⁡(5x2+25x2+42)5\frac{\log{\left(\frac{5 x}{2} + \frac{\sqrt{25 x^{2} + 4}}{2} \right)}}{5}

  2. Add the constant of integration:

    log⁡(5x2+25x2+42)5+constant\frac{\log{\left(\frac{5 x}{2} + \frac{\sqrt{25 x^{2} + 4}}{2} \right)}}{5}+ \mathrm{constant}

The answer is:

log⁡(5x2+25x2+42)5+constant\frac{\log{\left(\frac{5 x}{2} + \frac{\sqrt{25 x^{2} + 4}}{2} \right)}}{5}+ \mathrm{constant}

The answer (Indefinite) [src] 
                                /     ___________      \
                                |    /         2       |
  /                             |   /      25*x     5*x|
 |                           log|  /   1 + -----  + ---|
 |         1                    \\/          4       2 /
 | 1*-------------- dx = C + ---------------------------
 |      ___________                       5             
 |     /     2                                          
 |   \/  25*x  + 4                                      
 |                                                      
/
asinh  (5 x2)5{{{\rm asinh}\; \left({{5\,x}\over{2}}\right)}\over{5}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.01.0
Plot the graph f(x)
The answer [src] 
asinh(5/2)
----------
    5
asinh  (52)5{{{\rm asinh}\; \left({{5}\over{2}}\right)}\over{5}} 
=
= 
asinh(5/2)
----------
    5
asinh⁡(52)5\frac{\operatorname{asinh}{\left(\frac{5}{2} \right)}}{5}
Simplify
Numerical answer [src] 
0.329446229274219
0.329446229274219
The graph
Integral of dx/Sqrt[25x^2+4] dx

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

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