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Resolución de integrales/ (A^2)*(t^2)*(cos(2*Pi*f*t+d))^2

Integral de (A^2)*(t^2)*(cos(2*Pi*f*t+d))^2 dt

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
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 |   2  2    2                 
 |  a *t *cos (2*pi*f*t + d) dt
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∫01a2t2cos⁡2(d+t2πf) dt\int\limits_{0}^{1} a^{2} t^{2} \cos^{2}{\left(d + t 2 \pi f \right)}\, dt 
Integral((a^2*t^2)*cos(((2*pi)*f)*t + d)^2, (t, 0, 1))
Respuesta (Indefinida) [src] 
                                     //   /                                /d         \                                                          3/d         \                                                                                                                                                                                               2  2  2    3/d         \                                                        2/d         \                                                        4/d         \                                                3  3  3    4/d         \                                              3  3  3    2/d         \                                               2  2  2    /d         \                   \            \
                                     ||   |                           3*tan|- + pi*f*t|                                                     3*tan |- + pi*f*t|                                                                                                                                 3  3  3                                                  24*pi *f *t *tan |- + pi*f*t|                                           18*pi*f*t*tan |- + pi*f*t|                                            3*pi*f*t*tan |- + pi*f*t|                                            8*pi *f *t *tan |- + pi*f*t|                                         16*pi *f *t *tan |- + pi*f*t|                                          24*pi *f *t *tan|- + pi*f*t|                   |            |
                                     || 2 |                                \2         /                                                           \2         /                                                         3*pi*f*t                                                            8*pi *f *t                                                                    \2         /                                                         \2         /                                                         \2         /                                                            \2         /                                                          \2         /                                                          \2         /                   |            |
  /                                  ||a *|- ------------------------------------------------------------------- + ------------------------------------------------------------------- + ------------------------------------------------------------------- + ------------------------------------------------------------------- - ------------------------------------------------------------------- - ------------------------------------------------------------------- + ------------------------------------------------------------------- + ------------------------------------------------------------------- + ------------------------------------------------------------------- + -------------------------------------------------------------------|  for f != 0|
 |                                   ||   |       3  3        3  3    4/d         \        3  3    2/d         \        3  3        3  3    4/d         \        3  3    2/d         \        3  3        3  3    4/d         \        3  3    2/d         \        3  3        3  3    4/d         \        3  3    2/d         \        3  3        3  3    4/d         \        3  3    2/d         \        3  3        3  3    4/d         \        3  3    2/d         \        3  3        3  3    4/d         \        3  3    2/d         \        3  3        3  3    4/d         \        3  3    2/d         \        3  3        3  3    4/d         \        3  3    2/d         \        3  3        3  3    4/d         \        3  3    2/d         \|            |
 |  2  2    2                        ||   |  48*pi *f  + 48*pi *f *tan |- + pi*f*t| + 96*pi *f *tan |- + pi*f*t|   48*pi *f  + 48*pi *f *tan |- + pi*f*t| + 96*pi *f *tan |- + pi*f*t|   48*pi *f  + 48*pi *f *tan |- + pi*f*t| + 96*pi *f *tan |- + pi*f*t|   48*pi *f  + 48*pi *f *tan |- + pi*f*t| + 96*pi *f *tan |- + pi*f*t|   48*pi *f  + 48*pi *f *tan |- + pi*f*t| + 96*pi *f *tan |- + pi*f*t|   48*pi *f  + 48*pi *f *tan |- + pi*f*t| + 96*pi *f *tan |- + pi*f*t|   48*pi *f  + 48*pi *f *tan |- + pi*f*t| + 96*pi *f *tan |- + pi*f*t|   48*pi *f  + 48*pi *f *tan |- + pi*f*t| + 96*pi *f *tan |- + pi*f*t|   48*pi *f  + 48*pi *f *tan |- + pi*f*t| + 96*pi *f *tan |- + pi*f*t|   48*pi *f  + 48*pi *f *tan |- + pi*f*t| + 96*pi *f *tan |- + pi*f*t||            |
 | a *t *cos (2*pi*f*t + d) dt = C + |<   \                            \2         /                 \2         /                             \2         /                 \2         /                             \2         /                 \2         /                             \2         /                 \2         /                             \2         /                 \2         /                             \2         /                 \2         /                             \2         /                 \2         /                             \2         /                 \2         /                             \2         /                 \2         /                             \2         /                 \2         //            |
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/                                    ||                                                                                                                                                                                                                                                                                                                                                          2  3    2                                                                                                                                                                                                                                                                                                                                                                         |
                                     ||                                                                                                                                                                                                                                                                                                                                                         a *t *cos (d)                                                                                                                                                                                                                                                                                                                                                                      |
                                     ||                                                                                                                                                                                                                                                                                                                                                         -------------                                                                                                                                                                                                                                                                                                                                                            otherwise |
                                     ||                                                                                                                                                                                                                                                                                                                                                               3                                                                                                                                                                                                                                                                                                                                                                            |
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∫a2t2cos⁡2(d+t2πf) dt=C+{a2(8π3f3t3tan⁡4(d2+πft)48π3f3tan⁡4(d2+πft)+96π3f3tan⁡2(d2+πft)+48π3f3+16π3f3t3tan⁡2(d2+πft)48π3f3tan⁡4(d2+πft)+96π3f3tan⁡2(d2+πft)+48π3f3+8π3f3t348π3f3tan⁡4(d2+πft)+96π3f3tan⁡2(d2+πft)+48π3f3−24π2f2t2tan⁡3(d2+πft)48π3f3tan⁡4(d2+πft)+96π3f3tan⁡2(d2+πft)+48π3f3+24π2f2t2tan⁡(d2+πft)48π3f3tan⁡4(d2+πft)+96π3f3tan⁡2(d2+πft)+48π3f3+3πfttan⁡4(d2+πft)48π3f3tan⁡4(d2+πft)+96π3f3tan⁡2(d2+πft)+48π3f3−18πfttan⁡2(d2+πft)48π3f3tan⁡4(d2+πft)+96π3f3tan⁡2(d2+πft)+48π3f3+3πft48π3f3tan⁡4(d2+πft)+96π3f3tan⁡2(d2+πft)+48π3f3+3tan⁡3(d2+πft)48π3f3tan⁡4(d2+πft)+96π3f3tan⁡2(d2+πft)+48π3f3−3tan⁡(d2+πft)48π3f3tan⁡4(d2+πft)+96π3f3tan⁡2(d2+πft)+48π3f3)for f≠0a2t3cos⁡2(d)3otherwise\int a^{2} t^{2} \cos^{2}{\left(d + t 2 \pi f \right)}\, dt = C + \begin{cases} a^{2} \left(\frac{8 \pi^{3} f^{3} t^{3} \tan^{4}{\left(\frac{d}{2} + \pi f t \right)}}{48 \pi^{3} f^{3} \tan^{4}{\left(\frac{d}{2} + \pi f t \right)} + 96 \pi^{3} f^{3} \tan^{2}{\left(\frac{d}{2} + \pi f t \right)} + 48 \pi^{3} f^{3}} + \frac{16 \pi^{3} f^{3} t^{3} \tan^{2}{\left(\frac{d}{2} + \pi f t \right)}}{48 \pi^{3} f^{3} \tan^{4}{\left(\frac{d}{2} + \pi f t \right)} + 96 \pi^{3} f^{3} \tan^{2}{\left(\frac{d}{2} + \pi f t \right)} + 48 \pi^{3} f^{3}} + \frac{8 \pi^{3} f^{3} t^{3}}{48 \pi^{3} f^{3} \tan^{4}{\left(\frac{d}{2} + \pi f t \right)} + 96 \pi^{3} f^{3} \tan^{2}{\left(\frac{d}{2} + \pi f t \right)} + 48 \pi^{3} f^{3}} - \frac{24 \pi^{2} f^{2} t^{2} \tan^{3}{\left(\frac{d}{2} + \pi f t \right)}}{48 \pi^{3} f^{3} \tan^{4}{\left(\frac{d}{2} + \pi f t \right)} + 96 \pi^{3} f^{3} \tan^{2}{\left(\frac{d}{2} + \pi f t \right)} + 48 \pi^{3} f^{3}} + \frac{24 \pi^{2} f^{2} t^{2} \tan{\left(\frac{d}{2} + \pi f t \right)}}{48 \pi^{3} f^{3} \tan^{4}{\left(\frac{d}{2} + \pi f t \right)} + 96 \pi^{3} f^{3} \tan^{2}{\left(\frac{d}{2} + \pi f t \right)} + 48 \pi^{3} f^{3}} + \frac{3 \pi f t \tan^{4}{\left(\frac{d}{2} + \pi f t \right)}}{48 \pi^{3} f^{3} \tan^{4}{\left(\frac{d}{2} + \pi f t \right)} + 96 \pi^{3} f^{3} \tan^{2}{\left(\frac{d}{2} + \pi f t \right)} + 48 \pi^{3} f^{3}} - \frac{18 \pi f t \tan^{2}{\left(\frac{d}{2} + \pi f t \right)}}{48 \pi^{3} f^{3} \tan^{4}{\left(\frac{d}{2} + \pi f t \right)} + 96 \pi^{3} f^{3} \tan^{2}{\left(\frac{d}{2} + \pi f t \right)} + 48 \pi^{3} f^{3}} + \frac{3 \pi f t}{48 \pi^{3} f^{3} \tan^{4}{\left(\frac{d}{2} + \pi f t \right)} + 96 \pi^{3} f^{3} \tan^{2}{\left(\frac{d}{2} + \pi f t \right)} + 48 \pi^{3} f^{3}} + \frac{3 \tan^{3}{\left(\frac{d}{2} + \pi f t \right)}}{48 \pi^{3} f^{3} \tan^{4}{\left(\frac{d}{2} + \pi f t \right)} + 96 \pi^{3} f^{3} \tan^{2}{\left(\frac{d}{2} + \pi f t \right)} + 48 \pi^{3} f^{3}} - \frac{3 \tan{\left(\frac{d}{2} + \pi f t \right)}}{48 \pi^{3} f^{3} \tan^{4}{\left(\frac{d}{2} + \pi f t \right)} + 96 \pi^{3} f^{3} \tan^{2}{\left(\frac{d}{2} + \pi f t \right)} + 48 \pi^{3} f^{3}}\right) & \text{for}\: f \neq 0 \\\frac{a^{2} t^{3} \cos^{2}{\left(d \right)}}{3} & \text{otherwise} \end{cases}
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Respuesta [src] 
/   /   2                  2                  2                  2                                                                                \    2                                                
| 2 |cos (d + 2*pi*f)   sin (d + 2*pi*f)   sin (d + 2*pi*f)   cos (d + 2*pi*f)   cos(d + 2*pi*f)*sin(d + 2*pi*f)   cos(d + 2*pi*f)*sin(d + 2*pi*f)|   a *cos(d)*sin(d)                                  
|a *|---------------- + ---------------- - ---------------- + ---------------- - ------------------------------- + -------------------------------| + ----------------  for And(f > -oo, f < oo, f != 0)
|   |       6                  6                   2  2               2  2                       3  3                           4*pi*f            |           3  3                                      
|   \                                         16*pi *f           16*pi *f                   32*pi *f                                              /      32*pi *f                                       
<                                                                                                                                                                                                       
|                                                                              2    2                                                                                                                   
|                                                                             a *cos (d)                                                                                                                
|                                                                             ----------                                                                                           otherwise            
|                                                                                 3                                                                                                                     
\
{a2(sin⁡2(d+2πf)6+cos⁡2(d+2πf)6+sin⁡(d+2πf)cos⁡(d+2πf)4πf−sin⁡2(d+2πf)16π2f2+cos⁡2(d+2πf)16π2f2−sin⁡(d+2πf)cos⁡(d+2πf)32π3f3)+a2sin⁡(d)cos⁡(d)32π3f3for f>−∞∧f<∞∧f≠0a2cos⁡2(d)3otherwise\begin{cases} a^{2} \left(\frac{\sin^{2}{\left(d + 2 \pi f \right)}}{6} + \frac{\cos^{2}{\left(d + 2 \pi f \right)}}{6} + \frac{\sin{\left(d + 2 \pi f \right)} \cos{\left(d + 2 \pi f \right)}}{4 \pi f} - \frac{\sin^{2}{\left(d + 2 \pi f \right)}}{16 \pi^{2} f^{2}} + \frac{\cos^{2}{\left(d + 2 \pi f \right)}}{16 \pi^{2} f^{2}} - \frac{\sin{\left(d + 2 \pi f \right)} \cos{\left(d + 2 \pi f \right)}}{32 \pi^{3} f^{3}}\right) + \frac{a^{2} \sin{\left(d \right)} \cos{\left(d \right)}}{32 \pi^{3} f^{3}} & \text{for}\: f > -\infty \wedge f < \infty \wedge f \neq 0 \\\frac{a^{2} \cos^{2}{\left(d \right)}}{3} & \text{otherwise} \end{cases} 
=
= 
/   /   2                  2                  2                  2                                                                                \    2                                                
| 2 |cos (d + 2*pi*f)   sin (d + 2*pi*f)   sin (d + 2*pi*f)   cos (d + 2*pi*f)   cos(d + 2*pi*f)*sin(d + 2*pi*f)   cos(d + 2*pi*f)*sin(d + 2*pi*f)|   a *cos(d)*sin(d)                                  
|a *|---------------- + ---------------- - ---------------- + ---------------- - ------------------------------- + -------------------------------| + ----------------  for And(f > -oo, f < oo, f != 0)
|   |       6                  6                   2  2               2  2                       3  3                           4*pi*f            |           3  3                                      
|   \                                         16*pi *f           16*pi *f                   32*pi *f                                              /      32*pi *f                                       
<                                                                                                                                                                                                       
|                                                                              2    2                                                                                                                   
|                                                                             a *cos (d)                                                                                                                
|                                                                             ----------                                                                                           otherwise            
|                                                                                 3                                                                                                                     
\
{a2(sin⁡2(d+2πf)6+cos⁡2(d+2πf)6+sin⁡(d+2πf)cos⁡(d+2πf)4πf−sin⁡2(d+2πf)16π2f2+cos⁡2(d+2πf)16π2f2−sin⁡(d+2πf)cos⁡(d+2πf)32π3f3)+a2sin⁡(d)cos⁡(d)32π3f3for f>−∞∧f<∞∧f≠0a2cos⁡2(d)3otherwise\begin{cases} a^{2} \left(\frac{\sin^{2}{\left(d + 2 \pi f \right)}}{6} + \frac{\cos^{2}{\left(d + 2 \pi f \right)}}{6} + \frac{\sin{\left(d + 2 \pi f \right)} \cos{\left(d + 2 \pi f \right)}}{4 \pi f} - \frac{\sin^{2}{\left(d + 2 \pi f \right)}}{16 \pi^{2} f^{2}} + \frac{\cos^{2}{\left(d + 2 \pi f \right)}}{16 \pi^{2} f^{2}} - \frac{\sin{\left(d + 2 \pi f \right)} \cos{\left(d + 2 \pi f \right)}}{32 \pi^{3} f^{3}}\right) + \frac{a^{2} \sin{\left(d \right)} \cos{\left(d \right)}}{32 \pi^{3} f^{3}} & \text{for}\: f > -\infty \wedge f < \infty \wedge f \neq 0 \\\frac{a^{2} \cos^{2}{\left(d \right)}}{3} & \text{otherwise} \end{cases} 
Piecewise((a^2*(cos(d + 2*pi*f)^2/6 + sin(d + 2*pi*f)^2/6 - sin(d + 2*pi*f)^2/(16*pi^2*f^2) + cos(d + 2*pi*f)^2/(16*pi^2*f^2) - cos(d + 2*pi*f)*sin(d + 2*pi*f)/(32*pi^3*f^3) + cos(d + 2*pi*f)*sin(d + 2*pi*f)/(4*pi*f)) + a^2*cos(d)*sin(d)/(32*pi^3*f^3), (f > -oo)∧(f < oo)∧(Ne(f, 0))), (a^2*cos(d)^2/3, True))

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

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