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Solving integrals/ (dh)/(sqrt(2g*h))

Integral of (dh)/(sqrt(2g*h)) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1               
  /               
 |                
 |        1       
 |  1*--------- dh
 |      _______   
 |    \/ 2*g*h    
 |                
/                 
0
∫011⋅12gh dh\int\limits_{0}^{1} 1 \cdot \frac{1}{\sqrt{2 g h}}\, dh 
Integral(1/sqrt(2*g*h), (h, 0, 1))
Detail solution

    PiecewiseRule(subfunctions=[(URule(u_var=_u, u_func=sqrt(2)*sqrt(g*h), constant=None, substep=ConstantTimesRule(constant=1/g, other=_u/sqrt(_u**2), substep=ConstantTimesRule(constant=1/2, other=2*_u/sqrt(_u**2), substep=URule(u_var=_u, u_func=_u**2, constant=None, substep=PowerRule(base=_u, exp=-1/2, context=1/sqrt(_u), symbol=_u), context=2*_u/sqrt(_u**2), symbol=_u), context=_u/sqrt(_u**2), symbol=_u), context=_u/(g*sqrt(_u**2)), symbol=_u), context=1/sqrt(2*g*h), symbol=h), Ne(2*g, 0)), (ConstantRule(constant=zoo, context=zoo, symbol=h), True)], context=1/sqrt(2*g*h), symbol=h)

  1. Now simplify:

    {2ghgfor g>0∨g<0∞~hotherwise\begin{cases} \frac{\sqrt{2} \sqrt{g h}}{g} & \text{for}\: g > 0 \vee g < 0 \\\tilde{\infty} h & \text{otherwise} \end{cases}

  2. Add the constant of integration:

    {2ghgfor g>0∨g<0∞~hotherwise+constant\begin{cases} \frac{\sqrt{2} \sqrt{g h}}{g} & \text{for}\: g > 0 \vee g < 0 \\\tilde{\infty} h & \text{otherwise} \end{cases}+ \mathrm{constant}

The answer is:

{2ghgfor g>0∨g<0∞~hotherwise+constant\begin{cases} \frac{\sqrt{2} \sqrt{g h}}{g} & \text{for}\: g > 0 \vee g < 0 \\\tilde{\infty} h & \text{otherwise} \end{cases}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                     //  ___   _____            \
 |                      ||\/ 2 *\/ g*h             |
 |       1              ||-------------  for g != 0|
 | 1*--------- dh = C + |<      g                  |
 |     _______          ||                         |
 |   \/ 2*g*h           ||    zoo*h      otherwise |
 |                      \\                         /
/
∫1⋅12gh dh=C+{2ghgfor g≠0∞~hotherwise\int 1 \cdot \frac{1}{\sqrt{2 g h}}\, dh = C + \begin{cases} \frac{\sqrt{2} \sqrt{g h}}{g} & \text{for}\: g \neq 0 \\\tilde{\infty} h & \text{otherwise} \end{cases}
Simplify
The answer [src] 
  ___
\/ 2 
-----
  ___
\/ g
2g\frac{\sqrt{2}}{\sqrt{g}} 
=
= 
  ___
\/ 2 
-----
  ___
\/ g
2g\frac{\sqrt{2}}{\sqrt{g}}
Упростить

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

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