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Resolución de integrales/ (3x^2)/ (sqrt13)/((3x^2)-7)

Integral de (sqrt13)/((3x^2)-7) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1            
  /            
 |             
 |     ____    
 |   \/ 13     
 |  -------- dx
 |     2       
 |  3*x  - 7   
 |             
/              
0
∫01133x2−7 dx\int\limits_{0}^{1} \frac{\sqrt{13}}{3 x^{2} - 7}\, dx 
Integral(sqrt(13)/(3*x^2 - 7), (x, 0, 1))
Solución detallada

  1. La integral del producto de una función por una constante es la constante por la integral de esta función:

    ∫133x2−7 dx=13∫13x2−7 dx\int \frac{\sqrt{13}}{3 x^{2} - 7}\, dx = \sqrt{13} \int \frac{1}{3 x^{2} - 7}\, dx

      PieceweseRule(subfunctions=[(ArctanRule(a=1, b=3, c=-7, context=1/(3*x**2 - 7), symbol=x), False), (ArccothRule(a=1, b=3, c=-7, context=1/(3*x**2 - 7), symbol=x), x**2 > 7/3), (ArctanhRule(a=1, b=3, c=-7, context=1/(3*x**2 - 7), symbol=x), x**2 < 7/3)], context=1/(3*x**2 - 7), symbol=x)

    Por lo tanto, el resultado es: 13({−21acoth⁡(21x7)21for x2>73−21atanh⁡(21x7)21for x2<73)\sqrt{13} \left(\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left(\frac{\sqrt{21} x}{7} \right)}}{21} & \text{for}\: x^{2} > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left(\frac{\sqrt{21} x}{7} \right)}}{21} & \text{for}\: x^{2} < \frac{7}{3} \end{cases}\right)

  2. Ahora simplificar:

    {−273acoth⁡(21x7)21for x2>73−273atanh⁡(21x7)21for x2<73\begin{cases} - \frac{\sqrt{273} \operatorname{acoth}{\left(\frac{\sqrt{21} x}{7} \right)}}{21} & \text{for}\: x^{2} > \frac{7}{3} \\- \frac{\sqrt{273} \operatorname{atanh}{\left(\frac{\sqrt{21} x}{7} \right)}}{21} & \text{for}\: x^{2} < \frac{7}{3} \end{cases}

  3. Añadimos la constante de integración:

    {−273acoth⁡(21x7)21for x2>73−273atanh⁡(21x7)21for x2<73+constant\begin{cases} - \frac{\sqrt{273} \operatorname{acoth}{\left(\frac{\sqrt{21} x}{7} \right)}}{21} & \text{for}\: x^{2} > \frac{7}{3} \\- \frac{\sqrt{273} \operatorname{atanh}{\left(\frac{\sqrt{21} x}{7} \right)}}{21} & \text{for}\: x^{2} < \frac{7}{3} \end{cases}+ \mathrm{constant}

Respuesta:

{−273acoth⁡(21x7)21for x2>73−273atanh⁡(21x7)21for x2<73+constant\begin{cases} - \frac{\sqrt{273} \operatorname{acoth}{\left(\frac{\sqrt{21} x}{7} \right)}}{21} & \text{for}\: x^{2} > \frac{7}{3} \\- \frac{\sqrt{273} \operatorname{atanh}{\left(\frac{\sqrt{21} x}{7} \right)}}{21} & \text{for}\: x^{2} < \frac{7}{3} \end{cases}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                            //             /    ____\               \
                            ||   ____      |x*\/ 21 |               |
  /                         ||-\/ 21 *acoth|--------|               |
 |                          ||             \   7    /        2      |
 |    ____                  ||------------------------  for x  > 7/3|
 |  \/ 13              ____ ||           21                         |
 | -------- dx = C + \/ 13 *|<                                      |
 |    2                     ||             /    ____\               |
 | 3*x  - 7                 ||   ____      |x*\/ 21 |               |
 |                          ||-\/ 21 *atanh|--------|               |
/                           ||             \   7    /        2      |
                            ||------------------------  for x  < 7/3|
                            \\           21                         /
∫133x2−7 dx=C+13({−21acoth⁡(21x7)21for x2>73−21atanh⁡(21x7)21for x2<73)\int \frac{\sqrt{13}}{3 x^{2} - 7}\, dx = C + \sqrt{13} \left(\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left(\frac{\sqrt{21} x}{7} \right)}}{21} & \text{for}\: x^{2} > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left(\frac{\sqrt{21} x}{7} \right)}}{21} & \text{for}\: x^{2} < \frac{7}{3} \end{cases}\right)
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.90-1.0-0.5
Trazar el gráfico f(x)
Respuesta [src] 
       /            /      ____\          /          /       ____\\\          /            /  ____\          /          /  ____\\\
       |    ____    |    \/ 21 |     ____ |          |     \/ 21 |||          |    ____    |\/ 21 |     ____ |          |\/ 21 |||
       |  \/ 21 *log|1 + ------|   \/ 21 *|pi*I + log|-1 + ------|||          |  \/ 21 *log|------|   \/ 21 *|pi*I + log|------|||
  ____ |            \      3   /          \          \       3   //|     ____ |            \  3   /          \          \  3   //|
\/ 13 *|- ---------------------- + --------------------------------| - \/ 13 *|- ------------------ + ---------------------------|
       \            42                            42               /          \          42                        42            /
−13(−21log⁡(213)42+21(log⁡(213)+iπ)42)+13(−21log⁡(1+213)42+21(log⁡(−1+213)+iπ)42)- \sqrt{13} \left(- \frac{\sqrt{21} \log{\left(\frac{\sqrt{21}}{3} \right)}}{42} + \frac{\sqrt{21} \left(\log{\left(\frac{\sqrt{21}}{3} \right)} + i \pi\right)}{42}\right) + \sqrt{13} \left(- \frac{\sqrt{21} \log{\left(1 + \frac{\sqrt{21}}{3} \right)}}{42} + \frac{\sqrt{21} \left(\log{\left(-1 + \frac{\sqrt{21}}{3} \right)} + i \pi\right)}{42}\right) 
=
= 
       /            /      ____\          /          /       ____\\\          /            /  ____\          /          /  ____\\\
       |    ____    |    \/ 21 |     ____ |          |     \/ 21 |||          |    ____    |\/ 21 |     ____ |          |\/ 21 |||
       |  \/ 21 *log|1 + ------|   \/ 21 *|pi*I + log|-1 + ------|||          |  \/ 21 *log|------|   \/ 21 *|pi*I + log|------|||
  ____ |            \      3   /          \          \       3   //|     ____ |            \  3   /          \          \  3   //|
\/ 13 *|- ---------------------- + --------------------------------| - \/ 13 *|- ------------------ + ---------------------------|
       \            42                            42               /          \          42                        42            /
−13(−21log⁡(213)42+21(log⁡(213)+iπ)42)+13(−21log⁡(1+213)42+21(log⁡(−1+213)+iπ)42)- \sqrt{13} \left(- \frac{\sqrt{21} \log{\left(\frac{\sqrt{21}}{3} \right)}}{42} + \frac{\sqrt{21} \left(\log{\left(\frac{\sqrt{21}}{3} \right)} + i \pi\right)}{42}\right) + \sqrt{13} \left(- \frac{\sqrt{21} \log{\left(1 + \frac{\sqrt{21}}{3} \right)}}{42} + \frac{\sqrt{21} \left(\log{\left(-1 + \frac{\sqrt{21}}{3} \right)} + i \pi\right)}{42}\right) 
sqrt(13)*(-sqrt(21)*log(1 + sqrt(21)/3)/42 + sqrt(21)*(pi*i + log(-1 + sqrt(21)/3))/42) - sqrt(13)*(-sqrt(21)*log(sqrt(21)/3)/42 + sqrt(21)*(pi*i + log(sqrt(21)/3))/42)
Respuesta numérica [src] 
-0.616375523647113
-0.616375523647113

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

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