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Resolución de integrales/ (3x^2)/ dx/((3x^2)-5^2)

Integral de dx/((3x^2)-5^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1             
  /             
 |              
 |      1       
 |  --------- dx
 |     2        
 |  3*x  - 25   
 |              
/               
0
∫0113x2−25 dx\int\limits_{0}^{1} \frac{1}{3 x^{2} - 25}\, dx 
Integral(1/(3*x^2 - 25), (x, 0, 1))
Solución detallada

    PieceweseRule(subfunctions=[(ArctanRule(a=1, b=3, c=-25, context=1/(3*x**2 - 25), symbol=x), False), (ArccothRule(a=1, b=3, c=-25, context=1/(3*x**2 - 25), symbol=x), x**2 > 25/3), (ArctanhRule(a=1, b=3, c=-25, context=1/(3*x**2 - 25), symbol=x), x**2 < 25/3)], context=1/(3*x**2 - 25), symbol=x)

  1. Añadimos la constante de integración:

    {−3acoth⁡(3x5)15for x2>253−3atanh⁡(3x5)15for x2<253+constant\begin{cases} - \frac{\sqrt{3} \operatorname{acoth}{\left(\frac{\sqrt{3} x}{5} \right)}}{15} & \text{for}\: x^{2} > \frac{25}{3} \\- \frac{\sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3} x}{5} \right)}}{15} & \text{for}\: x^{2} < \frac{25}{3} \end{cases}+ \mathrm{constant}

Respuesta:

{−3acoth⁡(3x5)15for x2>253−3atanh⁡(3x5)15for x2<253+constant\begin{cases} - \frac{\sqrt{3} \operatorname{acoth}{\left(\frac{\sqrt{3} x}{5} \right)}}{15} & \text{for}\: x^{2} > \frac{25}{3} \\- \frac{\sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3} x}{5} \right)}}{15} & \text{for}\: x^{2} < \frac{25}{3} \end{cases}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                      //            /    ___\                \
                      ||   ___      |x*\/ 3 |                |
                      ||-\/ 3 *acoth|-------|                |
  /                   ||            \   5   /        2       |
 |                    ||----------------------  for x  > 25/3|
 |     1              ||          15                         |
 | --------- dx = C + |<                                     |
 |    2               ||            /    ___\                |
 | 3*x  - 25          ||   ___      |x*\/ 3 |                |
 |                    ||-\/ 3 *atanh|-------|                |
/                     ||            \   5   /        2       |
                      ||----------------------  for x  < 25/3|
                      \\          15                         /
∫13x2−25 dx=C+{−3acoth⁡(3x5)15for x2>253−3atanh⁡(3x5)15for x2<253\int \frac{1}{3 x^{2} - 25}\, dx = C + \begin{cases} - \frac{\sqrt{3} \operatorname{acoth}{\left(\frac{\sqrt{3} x}{5} \right)}}{15} & \text{for}\: x^{2} > \frac{25}{3} \\- \frac{\sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3} x}{5} \right)}}{15} & \text{for}\: x^{2} < \frac{25}{3} \end{cases}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.90-0.050-0.035
Trazar el gráfico f(x)
Respuesta [src] 
        /          /    ___\\            /        ___\         /          /         ___\\            /    ___\
    ___ |          |5*\/ 3 ||     ___    |    5*\/ 3 |     ___ |          |     5*\/ 3 ||     ___    |5*\/ 3 |
  \/ 3 *|pi*I + log|-------||   \/ 3 *log|1 + -------|   \/ 3 *|pi*I + log|-1 + -------||   \/ 3 *log|-------|
        \          \   3   //            \       3   /         \          \        3   //            \   3   /
- --------------------------- - ---------------------- + -------------------------------- + ------------------
               30                         30                            30                          30
−3log⁡(1+533)30+3log⁡(533)30−3(log⁡(533)+iπ)30+3(log⁡(−1+533)+iπ)30- \frac{\sqrt{3} \log{\left(1 + \frac{5 \sqrt{3}}{3} \right)}}{30} + \frac{\sqrt{3} \log{\left(\frac{5 \sqrt{3}}{3} \right)}}{30} - \frac{\sqrt{3} \left(\log{\left(\frac{5 \sqrt{3}}{3} \right)} + i \pi\right)}{30} + \frac{\sqrt{3} \left(\log{\left(-1 + \frac{5 \sqrt{3}}{3} \right)} + i \pi\right)}{30} 
=
= 
        /          /    ___\\            /        ___\         /          /         ___\\            /    ___\
    ___ |          |5*\/ 3 ||     ___    |    5*\/ 3 |     ___ |          |     5*\/ 3 ||     ___    |5*\/ 3 |
  \/ 3 *|pi*I + log|-------||   \/ 3 *log|1 + -------|   \/ 3 *|pi*I + log|-1 + -------||   \/ 3 *log|-------|
        \          \   3   //            \       3   /         \          \        3   //            \   3   /
- --------------------------- - ---------------------- + -------------------------------- + ------------------
               30                         30                            30                          30
−3log⁡(1+533)30+3log⁡(533)30−3(log⁡(533)+iπ)30+3(log⁡(−1+533)+iπ)30- \frac{\sqrt{3} \log{\left(1 + \frac{5 \sqrt{3}}{3} \right)}}{30} + \frac{\sqrt{3} \log{\left(\frac{5 \sqrt{3}}{3} \right)}}{30} - \frac{\sqrt{3} \left(\log{\left(\frac{5 \sqrt{3}}{3} \right)} + i \pi\right)}{30} + \frac{\sqrt{3} \left(\log{\left(-1 + \frac{5 \sqrt{3}}{3} \right)} + i \pi\right)}{30} 
-sqrt(3)*(pi*i + log(5*sqrt(3)/3))/30 - sqrt(3)*log(1 + 5*sqrt(3)/3)/30 + sqrt(3)*(pi*i + log(-1 + 5*sqrt(3)/3))/30 + sqrt(3)*log(5*sqrt(3)/3)/30
Respuesta numérica [src] 
-0.0417260966265954
-0.0417260966265954

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

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