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Resolución de integrales/ (3x+1)/ 1/(2x+sqr(3x+1))

Integral de 1/(2x+sqr(3x+1)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  0                    
  /                    
 |                     
 |         1           
 |  ---------------- dx
 |                 2   
 |  2*x + (3*x + 1)    
 |                     
/                      
5
$$\int\limits_{5}^{0} \frac{1}{2 x + \left(3 x + 1\right)^{2}}\, dx$$ 
Integral(1/(2*x + (3*x + 1)^2), (x, 5, 0))
Respuesta (Indefinida) [src] 
                               //            /    ___          \                        \
                               ||   ___      |9*\/ 7 *(4/9 + x)|                        |
                               ||-\/ 7 *acoth|-----------------|                        |
  /                            ||            \        7        /                2       |
 |                             ||--------------------------------  for (4/9 + x)  > 7/81|
 |        1                    ||               63                                      |
 | ---------------- dx = C + 9*|<                                                       |
 |                2            ||            /    ___          \                        |
 | 2*x + (3*x + 1)             ||   ___      |9*\/ 7 *(4/9 + x)|                        |
 |                             ||-\/ 7 *atanh|-----------------|                        |
/                              ||            \        7        /                2       |
                               ||--------------------------------  for (4/9 + x)  < 7/81|
                               \\               63                                      /
$$\int \frac{1}{2 x + \left(3 x + 1\right)^{2}}\, dx = C + 9 \left(\begin{cases} - \frac{\sqrt{7} \operatorname{acoth}{\left(\frac{9 \sqrt{7} \left(x + \frac{4}{9}\right)}{7} \right)}}{63} & \text{for}\: \left(x + \frac{4}{9}\right)^{2} > \frac{7}{81} \\- \frac{\sqrt{7} \operatorname{atanh}{\left(\frac{9 \sqrt{7} \left(x + \frac{4}{9}\right)}{7} \right)}}{63} & \text{for}\: \left(x + \frac{4}{9}\right)^{2} < \frac{7}{81} \end{cases}\right)$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
           /      ___\            /       ___\            /      ___\            /       ___\
    ___    |4   \/ 7 |     ___    |49   \/ 7 |     ___    |4   \/ 7 |     ___    |49   \/ 7 |
  \/ 7 *log|- + -----|   \/ 7 *log|-- - -----|   \/ 7 *log|- - -----|   \/ 7 *log|-- + -----|
           \9     9  /            \9      9  /            \9     9  /            \9      9  /
- -------------------- - --------------------- + -------------------- + ---------------------
           14                      14                     14                      14
$$\frac{\sqrt{7} \log{\left(\frac{4}{9} - \frac{\sqrt{7}}{9} \right)}}{14} - \frac{\sqrt{7} \log{\left(\frac{49}{9} - \frac{\sqrt{7}}{9} \right)}}{14} - \frac{\sqrt{7} \log{\left(\frac{\sqrt{7}}{9} + \frac{4}{9} \right)}}{14} + \frac{\sqrt{7} \log{\left(\frac{\sqrt{7}}{9} + \frac{49}{9} \right)}}{14}$$ 
=
= 
           /      ___\            /       ___\            /      ___\            /       ___\
    ___    |4   \/ 7 |     ___    |49   \/ 7 |     ___    |4   \/ 7 |     ___    |49   \/ 7 |
  \/ 7 *log|- + -----|   \/ 7 *log|-- - -----|   \/ 7 *log|- - -----|   \/ 7 *log|-- + -----|
           \9     9  /            \9      9  /            \9     9  /            \9      9  /
- -------------------- - --------------------- + -------------------- + ---------------------
           14                      14                     14                      14
$$\frac{\sqrt{7} \log{\left(\frac{4}{9} - \frac{\sqrt{7}}{9} \right)}}{14} - \frac{\sqrt{7} \log{\left(\frac{49}{9} - \frac{\sqrt{7}}{9} \right)}}{14} - \frac{\sqrt{7} \log{\left(\frac{\sqrt{7}}{9} + \frac{4}{9} \right)}}{14} + \frac{\sqrt{7} \log{\left(\frac{\sqrt{7}}{9} + \frac{49}{9} \right)}}{14}$$ 
-sqrt(7)*log(4/9 + sqrt(7)/9)/14 - sqrt(7)*log(49/9 - sqrt(7)/9)/14 + sqrt(7)*log(4/9 - sqrt(7)/9)/14 + sqrt(7)*log(49/9 + sqrt(7)/9)/14
Respuesta numérica [src] 
-0.280191856364123
-0.280191856364123

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

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