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Resolución de integrales/ log((t+h)/t)

Integral de log((t+h)/t) dt

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 h + t             
   /               
  |                
  |      /t + h\   
  |   log|-----| dt
  |      \  t  /   
  |                
 /                 
 t
∫th+tlog⁡(h+tt) dt\int\limits_{t}^{h + t} \log{\left(\frac{h + t}{t} \right)}\, dt 
Integral(log((t + h)/t), (t, t, h + t))
Respuesta (Indefinida) [src] 
                                        /      //       1    1                    \       //         1    1                   \\
  /                                     |      ||       - + ---          for h = 0|       ||       - - - ---         for h = 0||
 |                                      |      ||       t   2*h                   |       ||         t   2*h                  ||
 |    /t + h\               /t + h\     |      ||                                 |       ||                                  ||
 | log|-----| dt = C + t*log|-----| - h*|- 2*h*|<   /        /1    1 \\           | + 2*h*|<   /         /1    1 \\           ||
 |    \  t  /               \  t  /     |      ||log|1 + 2*h*|- + ---||           |       ||log|-1 + 2*h*|- + ---||           ||
 |                                      |      ||   \        \t   2*h//           |       ||   \         \t   2*h//           ||
/                                       |      ||----------------------  otherwise|       ||-----------------------  otherwise||
                                        \      \\         2*h                     /       \\          2*h                     //
∫log⁡(h+tt) dt=C−h(2h({−1t−12hfor h=0log⁡(2h(1t+12h)−1)2hotherwise)−2h({1t+12hfor h=0log⁡(2h(1t+12h)+1)2hotherwise))+tlog⁡(h+tt)\int \log{\left(\frac{h + t}{t} \right)}\, dt = C - h \left(2 h \left(\begin{cases} - \frac{1}{t} - \frac{1}{2 h} & \text{for}\: h = 0 \\\frac{\log{\left(2 h \left(\frac{1}{t} + \frac{1}{2 h}\right) - 1 \right)}}{2 h} & \text{otherwise} \end{cases}\right) - 2 h \left(\begin{cases} \frac{1}{t} + \frac{1}{2 h} & \text{for}\: h = 0 \\\frac{\log{\left(2 h \left(\frac{1}{t} + \frac{1}{2 h}\right) + 1 \right)}}{2 h} & \text{otherwise} \end{cases}\right)\right) + t \log{\left(\frac{h + t}{t} \right)}
Simplificar
Respuesta [src] 
                            /t + 2*h\                       /h + t\
h*log(t + 2*h) + (h + t)*log|-------| - h*log(h + t) - t*log|-----|
                            \ h + t /                       \  t  /
−hlog⁡(h+t)+hlog⁡(2h+t)−tlog⁡(h+tt)+(h+t)log⁡(2h+th+t)- h \log{\left(h + t \right)} + h \log{\left(2 h + t \right)} - t \log{\left(\frac{h + t}{t} \right)} + \left(h + t\right) \log{\left(\frac{2 h + t}{h + t} \right)} 
=
= 
                            /t + 2*h\                       /h + t\
h*log(t + 2*h) + (h + t)*log|-------| - h*log(h + t) - t*log|-----|
                            \ h + t /                       \  t  /
−hlog⁡(h+t)+hlog⁡(2h+t)−tlog⁡(h+tt)+(h+t)log⁡(2h+th+t)- h \log{\left(h + t \right)} + h \log{\left(2 h + t \right)} - t \log{\left(\frac{h + t}{t} \right)} + \left(h + t\right) \log{\left(\frac{2 h + t}{h + t} \right)} 
h*log(t + 2*h) + (h + t)*log((t + 2*h)/(h + t)) - h*log(h + t) - t*log((h + t)/t)

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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