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Resolución de integrales/ ln(4(pi^(2)+4y^(2))/pi^(2)+16y^2)

Integral de ln(4(pi^(2)+4y^(2))/pi^(2)+16y^2) dy

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                               
  /                               
 |                                
 |     /  /  2      2\        \   
 |     |4*\pi  + 4*y /       2|   
 |  log|-------------- + 16*y | dy
 |     |       2              |   
 |     \     pi               /   
 |                                
/                                 
0
∫01log⁡(16y2+4(4y2+π2)π2) dy\int\limits_{0}^{1} \log{\left(16 y^{2} + \frac{4 \left(4 y^{2} + \pi^{2}\right)}{\pi^{2}} \right)}\, dy 
Integral(log((4*(pi^2 + 4*y^2))/pi^2 + 16*y^2), (y, 0, 1))
Respuesta (Indefinida) [src] 
                                                                                       /     ___________\
  /                                                                                    |    /         2 |
 |                                                                                     |y*\/  4 + 4*pi  |
 |    /  /  2      2\        \                     /  /  2      2\        \   2*pi*atan|----------------|
 |    |4*\pi  + 4*y /       2|                     |4*\pi  + 4*y /       2|            \       pi       /
 | log|-------------- + 16*y | dy = C - 2*y + y*log|-------------- + 16*y | + ---------------------------
 |    |       2              |                     |       2              |             ___________      
 |    \     pi               /                     \     pi               /            /         2       
 |                                                                                   \/  4 + 4*pi        
/
∫log⁡(16y2+4(4y2+π2)π2) dy=C+ylog⁡(16y2+4(4y2+π2)π2)−2y+2πatan⁡(y4+4π2π)4+4π2\int \log{\left(16 y^{2} + \frac{4 \left(4 y^{2} + \pi^{2}\right)}{\pi^{2}} \right)}\, dy = C + y \log{\left(16 y^{2} + \frac{4 \left(4 y^{2} + \pi^{2}\right)}{\pi^{2}} \right)} - 2 y + \frac{2 \pi \operatorname{atan}{\left(\frac{y \sqrt{4 + 4 \pi^{2}}}{\pi} \right)}}{\sqrt{4 + 4 \pi^{2}}}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.9005
Trazar el gráfico f(x)
Respuesta [src] 
             /                   /     _________\\                       
             |                   |    /       2 ||                       
             |                   |2*\/  1 + pi  ||                       
             |            pi*atan|--------------||      /              2\
/         2\ |    1              \      pi      /|      |     16 + 4*pi |
\-8 - 8*pi /*|--------- - -----------------------| + log|16 + ----------|
             |        2                   3/2    |      |          2    |
             |4 + 4*pi           /      2\       |      \        pi     /
             \                 8*\1 + pi /       /
(−8π2−8)(−πatan⁡(21+π2π)8(1+π2)32+14+4π2)+log⁡(16+4π2π2+16)\left(- 8 \pi^{2} - 8\right) \left(- \frac{\pi \operatorname{atan}{\left(\frac{2 \sqrt{1 + \pi^{2}}}{\pi} \right)}}{8 \left(1 + \pi^{2}\right)^{\frac{3}{2}}} + \frac{1}{4 + 4 \pi^{2}}\right) + \log{\left(\frac{16 + 4 \pi^{2}}{\pi^{2}} + 16 \right)} 
=
= 
             /                   /     _________\\                       
             |                   |    /       2 ||                       
             |                   |2*\/  1 + pi  ||                       
             |            pi*atan|--------------||      /              2\
/         2\ |    1              \      pi      /|      |     16 + 4*pi |
\-8 - 8*pi /*|--------- - -----------------------| + log|16 + ----------|
             |        2                   3/2    |      |          2    |
             |4 + 4*pi           /      2\       |      \        pi     /
             \                 8*\1 + pi /       /
(−8π2−8)(−πatan⁡(21+π2π)8(1+π2)32+14+4π2)+log⁡(16+4π2π2+16)\left(- 8 \pi^{2} - 8\right) \left(- \frac{\pi \operatorname{atan}{\left(\frac{2 \sqrt{1 + \pi^{2}}}{\pi} \right)}}{8 \left(1 + \pi^{2}\right)^{\frac{3}{2}}} + \frac{1}{4 + 4 \pi^{2}}\right) + \log{\left(\frac{16 + 4 \pi^{2}}{\pi^{2}} + 16 \right)} 
(-8 - 8*pi^2)*(1/(4 + 4*pi^2) - pi*atan(2*sqrt(1 + pi^2)/pi)/(8*(1 + pi^2)^(3/2))) + log(16 + (16 + 4*pi^2)/pi^2)
Respuesta numérica [src] 
2.14678767771845
2.14678767771845

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

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