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Resolución de integrales/ cos(2.2*x)*cos((x-2.3)u)

Integral de cos(2.2*x)*cos((x-2.3)u) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 37                             
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 10                             
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 |     /11*x\    //    23\  \   
 |  cos|----|*cos||x - --|*u| dx
 |     \ 5  /    \\    10/  /   
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/                               
23                              
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10
∫23103710cos⁡(11x5)cos⁡(u(x−2310)) dx\int\limits_{\frac{23}{10}}^{\frac{37}{10}} \cos{\left(\frac{11 x}{5} \right)} \cos{\left(u \left(x - \frac{23}{10}\right) \right)}\, dx 
Integral(cos(11*x/5)*cos((x - 23/10)*u), (x, 23/10, 37/10))
Respuesta (Indefinida) [src] 
                                                                                                                                      //                        2/11*x\                                 \          
                                                                                                                                      ||                  -5*sin |----|                                 |          
                                                                                                                                      ||                         \ 5  /                                 |          
                                      //     2/11*x\        2/11*x\        /11*x\    /11*x\                             \             ||                  -------------                    for u = -11/5|          
                                      ||x*cos |----|   x*sin |----|   5*cos|----|*sin|----|                             |             ||                        22                                      |          
                                      ||      \ 5  /         \ 5  /        \ 5  /    \ 5  /                             |             ||                                                                |          
  /                                   ||------------ + ------------ + ---------------------  for Or(u = -11/5, u = 11/5)|             ||                       2/11*x\                                  |          
 |                                    ||     2              2                   22                                      |             ||                  5*sin |----|                                  |          
 |    /11*x\    //    23\  \          ||                                                                                |    /23*u\   ||                        \ 5  /                                  |    /23*u\
 | cos|----|*cos||x - --|*u| dx = C + |<                  /11*x\           /11*x\                                       |*cos|----| + |<                  ------------                     for u = 11/5 |*sin|----|
 |    \ 5  /    \\    10/  /          ||   55*cos(u*x)*sin|----|   25*u*cos|----|*sin(u*x)                              |    \ 10 /   ||                       22                                       |    \ 10 /
 |                                    ||                  \ 5  /           \ 5  /                                       |             ||                                                                |          
/                                     || - --------------------- + -----------------------            otherwise         |             ||        /11*x\                    /11*x\                        |          
                                      ||                   2                        2                                   |             ||  55*sin|----|*sin(u*x)   25*u*cos|----|*cos(u*x)               |          
                                      ||        -121 + 25*u              -121 + 25*u                                    |             ||        \ 5  /                    \ 5  /                        |          
                                      \\                                                                                /             ||- --------------------- - -----------------------    otherwise  |          
                                                                                                                                      ||                  2                        2                    |          
                                                                                                                                      ||       -121 + 25*u              -121 + 25*u                     |          
                                                                                                                                      \\                                                                /
∫cos⁡(11x5)cos⁡(u(x−2310)) dx=C+({xsin⁡2(11x5)2+xcos⁡2(11x5)2+5sin⁡(11x5)cos⁡(11x5)22for u=−115∨u=11525usin⁡(ux)cos⁡(11x5)25u2−121−55sin⁡(11x5)cos⁡(ux)25u2−121otherwise)cos⁡(23u10)+({−5sin⁡2(11x5)22for u=−1155sin⁡2(11x5)22for u=115−25ucos⁡(11x5)cos⁡(ux)25u2−121−55sin⁡(11x5)sin⁡(ux)25u2−121otherwise)sin⁡(23u10)\int \cos{\left(\frac{11 x}{5} \right)} \cos{\left(u \left(x - \frac{23}{10}\right) \right)}\, dx = C + \left(\begin{cases} \frac{x \sin^{2}{\left(\frac{11 x}{5} \right)}}{2} + \frac{x \cos^{2}{\left(\frac{11 x}{5} \right)}}{2} + \frac{5 \sin{\left(\frac{11 x}{5} \right)} \cos{\left(\frac{11 x}{5} \right)}}{22} & \text{for}\: u = - \frac{11}{5} \vee u = \frac{11}{5} \\\frac{25 u \sin{\left(u x \right)} \cos{\left(\frac{11 x}{5} \right)}}{25 u^{2} - 121} - \frac{55 \sin{\left(\frac{11 x}{5} \right)} \cos{\left(u x \right)}}{25 u^{2} - 121} & \text{otherwise} \end{cases}\right) \cos{\left(\frac{23 u}{10} \right)} + \left(\begin{cases} - \frac{5 \sin^{2}{\left(\frac{11 x}{5} \right)}}{22} & \text{for}\: u = - \frac{11}{5} \\\frac{5 \sin^{2}{\left(\frac{11 x}{5} \right)}}{22} & \text{for}\: u = \frac{11}{5} \\- \frac{25 u \cos{\left(\frac{11 x}{5} \right)} \cos{\left(u x \right)}}{25 u^{2} - 121} - \frac{55 \sin{\left(\frac{11 x}{5} \right)} \sin{\left(u x \right)}}{25 u^{2} - 121} & \text{otherwise} \end{cases}\right) \sin{\left(\frac{23 u}{10} \right)}
Simplificar
Respuesta [src] 
/                                        /      2/407\         2/407\        /407\    /407\\             /      2/253\         2/253\        /253\    /253\\                  2/407\    /23*u\        2/253\    /23*u\                                                       
|                                        |37*cos |---|   37*sin |---|   5*cos|---|*sin|---||             |23*cos |---|   23*sin |---|   5*cos|---|*sin|---||             5*sin |---|*sin|----|   5*sin |---|*sin|----|                                                       
|                                        |       \ 50/          \ 50/        \ 50/    \ 50/|    /23*u\   |       \ 50/          \ 50/        \ 50/    \ 50/|    /23*u\         \ 50/    \ 10 /         \ 50/    \ 10 /                                                       
|                                        |------------ + ------------ + -------------------|*cos|----| - |------------ + ------------ + -------------------|*cos|----| - --------------------- + ---------------------                                          for u = -11/5
|                                        \     20             20                 22        /    \ 10 /   \     20             20                 22        /    \ 10 /             22                      22                                                                
|                                                                                                                                                                                                                                                                            
|                                        /      2/407\         2/407\        /407\    /407\\             /      2/253\         2/253\        /253\    /253\\                  2/253\    /23*u\        2/407\    /23*u\                                                       
|                                        |37*cos |---|   37*sin |---|   5*cos|---|*sin|---||             |23*cos |---|   23*sin |---|   5*cos|---|*sin|---||             5*sin |---|*sin|----|   5*sin |---|*sin|----|                                                       
|                                        |       \ 50/          \ 50/        \ 50/    \ 50/|    /23*u\   |       \ 50/          \ 50/        \ 50/    \ 50/|    /23*u\         \ 50/    \ 10 /         \ 50/    \ 10 /                                                       
<                                        |------------ + ------------ + -------------------|*cos|----| - |------------ + ------------ + -------------------|*cos|----| - --------------------- + ---------------------                                          for u = 11/5 
|                                        \     20             20                 22        /    \ 10 /   \     20             20                 22        /    \ 10 /             22                      22                                                                
|                                                                                                                                                                                                                                                                            
|/        /37*u\    /407\           /407\    /37*u\\             /        /407\    /37*u\           /407\    /37*u\\             /        /23*u\    /253\           /253\    /23*u\\             /        /253\    /23*u\           /253\    /23*u\\                         
||  55*cos|----|*sin|---|   25*u*cos|---|*sin|----||             |  55*sin|---|*sin|----|   25*u*cos|---|*cos|----||             |  55*cos|----|*sin|---|   25*u*cos|---|*sin|----||             |  55*sin|---|*sin|----|   25*u*cos|---|*cos|----||                         
||        \ 10 /    \ 50/           \ 50/    \ 10 /|    /23*u\   |        \ 50/    \ 10 /           \ 50/    \ 10 /|    /23*u\   |        \ 10 /    \ 50/           \ 50/    \ 10 /|    /23*u\   |        \ 50/    \ 10 /           \ 50/    \ 10 /|    /23*u\               
||- --------------------- + -----------------------|*cos|----| + |- --------------------- - -----------------------|*sin|----| - |- --------------------- + -----------------------|*cos|----| - |- --------------------- - -----------------------|*sin|----|    otherwise  
||                  2                        2     |    \ 10 /   |                  2                        2     |    \ 10 /   |                  2                        2     |    \ 10 /   |                  2                        2     |    \ 10 /               
|\       -121 + 25*u              -121 + 25*u      /             \       -121 + 25*u              -121 + 25*u      /             \       -121 + 25*u              -121 + 25*u      /             \       -121 + 25*u              -121 + 25*u      /                         
\
{−5sin⁡2(40750)sin⁡(23u10)22+5sin⁡2(25350)sin⁡(23u10)22−(5sin⁡(25350)cos⁡(25350)22+23cos⁡2(25350)20+23sin⁡2(25350)20)cos⁡(23u10)+(5sin⁡(40750)cos⁡(40750)22+37cos⁡2(40750)20+37sin⁡2(40750)20)cos⁡(23u10)for u=−115−5sin⁡2(25350)sin⁡(23u10)22+5sin⁡2(40750)sin⁡(23u10)22−(5sin⁡(25350)cos⁡(25350)22+23cos⁡2(25350)20+23sin⁡2(25350)20)cos⁡(23u10)+(5sin⁡(40750)cos⁡(40750)22+37cos⁡2(40750)20+37sin⁡2(40750)20)cos⁡(23u10)for u=115−(25usin⁡(23u10)cos⁡(25350)25u2−121−55sin⁡(25350)cos⁡(23u10)25u2−121)cos⁡(23u10)+(25usin⁡(37u10)cos⁡(40750)25u2−121−55sin⁡(40750)cos⁡(37u10)25u2−121)cos⁡(23u10)−(−25ucos⁡(25350)cos⁡(23u10)25u2−121−55sin⁡(25350)sin⁡(23u10)25u2−121)sin⁡(23u10)+(−25ucos⁡(40750)cos⁡(37u10)25u2−121−55sin⁡(40750)sin⁡(37u10)25u2−121)sin⁡(23u10)otherwise\begin{cases} - \frac{5 \sin^{2}{\left(\frac{407}{50} \right)} \sin{\left(\frac{23 u}{10} \right)}}{22} + \frac{5 \sin^{2}{\left(\frac{253}{50} \right)} \sin{\left(\frac{23 u}{10} \right)}}{22} - \left(\frac{5 \sin{\left(\frac{253}{50} \right)} \cos{\left(\frac{253}{50} \right)}}{22} + \frac{23 \cos^{2}{\left(\frac{253}{50} \right)}}{20} + \frac{23 \sin^{2}{\left(\frac{253}{50} \right)}}{20}\right) \cos{\left(\frac{23 u}{10} \right)} + \left(\frac{5 \sin{\left(\frac{407}{50} \right)} \cos{\left(\frac{407}{50} \right)}}{22} + \frac{37 \cos^{2}{\left(\frac{407}{50} \right)}}{20} + \frac{37 \sin^{2}{\left(\frac{407}{50} \right)}}{20}\right) \cos{\left(\frac{23 u}{10} \right)} & \text{for}\: u = - \frac{11}{5} \\- \frac{5 \sin^{2}{\left(\frac{253}{50} \right)} \sin{\left(\frac{23 u}{10} \right)}}{22} + \frac{5 \sin^{2}{\left(\frac{407}{50} \right)} \sin{\left(\frac{23 u}{10} \right)}}{22} - \left(\frac{5 \sin{\left(\frac{253}{50} \right)} \cos{\left(\frac{253}{50} \right)}}{22} + \frac{23 \cos^{2}{\left(\frac{253}{50} \right)}}{20} + \frac{23 \sin^{2}{\left(\frac{253}{50} \right)}}{20}\right) \cos{\left(\frac{23 u}{10} \right)} + \left(\frac{5 \sin{\left(\frac{407}{50} \right)} \cos{\left(\frac{407}{50} \right)}}{22} + \frac{37 \cos^{2}{\left(\frac{407}{50} \right)}}{20} + \frac{37 \sin^{2}{\left(\frac{407}{50} \right)}}{20}\right) \cos{\left(\frac{23 u}{10} \right)} & \text{for}\: u = \frac{11}{5} \\- \left(\frac{25 u \sin{\left(\frac{23 u}{10} \right)} \cos{\left(\frac{253}{50} \right)}}{25 u^{2} - 121} - \frac{55 \sin{\left(\frac{253}{50} \right)} \cos{\left(\frac{23 u}{10} \right)}}{25 u^{2} - 121}\right) \cos{\left(\frac{23 u}{10} \right)} + \left(\frac{25 u \sin{\left(\frac{37 u}{10} \right)} \cos{\left(\frac{407}{50} \right)}}{25 u^{2} - 121} - \frac{55 \sin{\left(\frac{407}{50} \right)} \cos{\left(\frac{37 u}{10} \right)}}{25 u^{2} - 121}\right) \cos{\left(\frac{23 u}{10} \right)} - \left(- \frac{25 u \cos{\left(\frac{253}{50} \right)} \cos{\left(\frac{23 u}{10} \right)}}{25 u^{2} - 121} - \frac{55 \sin{\left(\frac{253}{50} \right)} \sin{\left(\frac{23 u}{10} \right)}}{25 u^{2} - 121}\right) \sin{\left(\frac{23 u}{10} \right)} + \left(- \frac{25 u \cos{\left(\frac{407}{50} \right)} \cos{\left(\frac{37 u}{10} \right)}}{25 u^{2} - 121} - \frac{55 \sin{\left(\frac{407}{50} \right)} \sin{\left(\frac{37 u}{10} \right)}}{25 u^{2} - 121}\right) \sin{\left(\frac{23 u}{10} \right)} & \text{otherwise} \end{cases} 
=
= 
/                                        /      2/407\         2/407\        /407\    /407\\             /      2/253\         2/253\        /253\    /253\\                  2/407\    /23*u\        2/253\    /23*u\                                                       
|                                        |37*cos |---|   37*sin |---|   5*cos|---|*sin|---||             |23*cos |---|   23*sin |---|   5*cos|---|*sin|---||             5*sin |---|*sin|----|   5*sin |---|*sin|----|                                                       
|                                        |       \ 50/          \ 50/        \ 50/    \ 50/|    /23*u\   |       \ 50/          \ 50/        \ 50/    \ 50/|    /23*u\         \ 50/    \ 10 /         \ 50/    \ 10 /                                                       
|                                        |------------ + ------------ + -------------------|*cos|----| - |------------ + ------------ + -------------------|*cos|----| - --------------------- + ---------------------                                          for u = -11/5
|                                        \     20             20                 22        /    \ 10 /   \     20             20                 22        /    \ 10 /             22                      22                                                                
|                                                                                                                                                                                                                                                                            
|                                        /      2/407\         2/407\        /407\    /407\\             /      2/253\         2/253\        /253\    /253\\                  2/253\    /23*u\        2/407\    /23*u\                                                       
|                                        |37*cos |---|   37*sin |---|   5*cos|---|*sin|---||             |23*cos |---|   23*sin |---|   5*cos|---|*sin|---||             5*sin |---|*sin|----|   5*sin |---|*sin|----|                                                       
|                                        |       \ 50/          \ 50/        \ 50/    \ 50/|    /23*u\   |       \ 50/          \ 50/        \ 50/    \ 50/|    /23*u\         \ 50/    \ 10 /         \ 50/    \ 10 /                                                       
<                                        |------------ + ------------ + -------------------|*cos|----| - |------------ + ------------ + -------------------|*cos|----| - --------------------- + ---------------------                                          for u = 11/5 
|                                        \     20             20                 22        /    \ 10 /   \     20             20                 22        /    \ 10 /             22                      22                                                                
|                                                                                                                                                                                                                                                                            
|/        /37*u\    /407\           /407\    /37*u\\             /        /407\    /37*u\           /407\    /37*u\\             /        /23*u\    /253\           /253\    /23*u\\             /        /253\    /23*u\           /253\    /23*u\\                         
||  55*cos|----|*sin|---|   25*u*cos|---|*sin|----||             |  55*sin|---|*sin|----|   25*u*cos|---|*cos|----||             |  55*cos|----|*sin|---|   25*u*cos|---|*sin|----||             |  55*sin|---|*sin|----|   25*u*cos|---|*cos|----||                         
||        \ 10 /    \ 50/           \ 50/    \ 10 /|    /23*u\   |        \ 50/    \ 10 /           \ 50/    \ 10 /|    /23*u\   |        \ 10 /    \ 50/           \ 50/    \ 10 /|    /23*u\   |        \ 50/    \ 10 /           \ 50/    \ 10 /|    /23*u\               
||- --------------------- + -----------------------|*cos|----| + |- --------------------- - -----------------------|*sin|----| - |- --------------------- + -----------------------|*cos|----| - |- --------------------- - -----------------------|*sin|----|    otherwise  
||                  2                        2     |    \ 10 /   |                  2                        2     |    \ 10 /   |                  2                        2     |    \ 10 /   |                  2                        2     |    \ 10 /               
|\       -121 + 25*u              -121 + 25*u      /             \       -121 + 25*u              -121 + 25*u      /             \       -121 + 25*u              -121 + 25*u      /             \       -121 + 25*u              -121 + 25*u      /                         
\
{−5sin⁡2(40750)sin⁡(23u10)22+5sin⁡2(25350)sin⁡(23u10)22−(5sin⁡(25350)cos⁡(25350)22+23cos⁡2(25350)20+23sin⁡2(25350)20)cos⁡(23u10)+(5sin⁡(40750)cos⁡(40750)22+37cos⁡2(40750)20+37sin⁡2(40750)20)cos⁡(23u10)for u=−115−5sin⁡2(25350)sin⁡(23u10)22+5sin⁡2(40750)sin⁡(23u10)22−(5sin⁡(25350)cos⁡(25350)22+23cos⁡2(25350)20+23sin⁡2(25350)20)cos⁡(23u10)+(5sin⁡(40750)cos⁡(40750)22+37cos⁡2(40750)20+37sin⁡2(40750)20)cos⁡(23u10)for u=115−(25usin⁡(23u10)cos⁡(25350)25u2−121−55sin⁡(25350)cos⁡(23u10)25u2−121)cos⁡(23u10)+(25usin⁡(37u10)cos⁡(40750)25u2−121−55sin⁡(40750)cos⁡(37u10)25u2−121)cos⁡(23u10)−(−25ucos⁡(25350)cos⁡(23u10)25u2−121−55sin⁡(25350)sin⁡(23u10)25u2−121)sin⁡(23u10)+(−25ucos⁡(40750)cos⁡(37u10)25u2−121−55sin⁡(40750)sin⁡(37u10)25u2−121)sin⁡(23u10)otherwise\begin{cases} - \frac{5 \sin^{2}{\left(\frac{407}{50} \right)} \sin{\left(\frac{23 u}{10} \right)}}{22} + \frac{5 \sin^{2}{\left(\frac{253}{50} \right)} \sin{\left(\frac{23 u}{10} \right)}}{22} - \left(\frac{5 \sin{\left(\frac{253}{50} \right)} \cos{\left(\frac{253}{50} \right)}}{22} + \frac{23 \cos^{2}{\left(\frac{253}{50} \right)}}{20} + \frac{23 \sin^{2}{\left(\frac{253}{50} \right)}}{20}\right) \cos{\left(\frac{23 u}{10} \right)} + \left(\frac{5 \sin{\left(\frac{407}{50} \right)} \cos{\left(\frac{407}{50} \right)}}{22} + \frac{37 \cos^{2}{\left(\frac{407}{50} \right)}}{20} + \frac{37 \sin^{2}{\left(\frac{407}{50} \right)}}{20}\right) \cos{\left(\frac{23 u}{10} \right)} & \text{for}\: u = - \frac{11}{5} \\- \frac{5 \sin^{2}{\left(\frac{253}{50} \right)} \sin{\left(\frac{23 u}{10} \right)}}{22} + \frac{5 \sin^{2}{\left(\frac{407}{50} \right)} \sin{\left(\frac{23 u}{10} \right)}}{22} - \left(\frac{5 \sin{\left(\frac{253}{50} \right)} \cos{\left(\frac{253}{50} \right)}}{22} + \frac{23 \cos^{2}{\left(\frac{253}{50} \right)}}{20} + \frac{23 \sin^{2}{\left(\frac{253}{50} \right)}}{20}\right) \cos{\left(\frac{23 u}{10} \right)} + \left(\frac{5 \sin{\left(\frac{407}{50} \right)} \cos{\left(\frac{407}{50} \right)}}{22} + \frac{37 \cos^{2}{\left(\frac{407}{50} \right)}}{20} + \frac{37 \sin^{2}{\left(\frac{407}{50} \right)}}{20}\right) \cos{\left(\frac{23 u}{10} \right)} & \text{for}\: u = \frac{11}{5} \\- \left(\frac{25 u \sin{\left(\frac{23 u}{10} \right)} \cos{\left(\frac{253}{50} \right)}}{25 u^{2} - 121} - \frac{55 \sin{\left(\frac{253}{50} \right)} \cos{\left(\frac{23 u}{10} \right)}}{25 u^{2} - 121}\right) \cos{\left(\frac{23 u}{10} \right)} + \left(\frac{25 u \sin{\left(\frac{37 u}{10} \right)} \cos{\left(\frac{407}{50} \right)}}{25 u^{2} - 121} - \frac{55 \sin{\left(\frac{407}{50} \right)} \cos{\left(\frac{37 u}{10} \right)}}{25 u^{2} - 121}\right) \cos{\left(\frac{23 u}{10} \right)} - \left(- \frac{25 u \cos{\left(\frac{253}{50} \right)} \cos{\left(\frac{23 u}{10} \right)}}{25 u^{2} - 121} - \frac{55 \sin{\left(\frac{253}{50} \right)} \sin{\left(\frac{23 u}{10} \right)}}{25 u^{2} - 121}\right) \sin{\left(\frac{23 u}{10} \right)} + \left(- \frac{25 u \cos{\left(\frac{407}{50} \right)} \cos{\left(\frac{37 u}{10} \right)}}{25 u^{2} - 121} - \frac{55 \sin{\left(\frac{407}{50} \right)} \sin{\left(\frac{37 u}{10} \right)}}{25 u^{2} - 121}\right) \sin{\left(\frac{23 u}{10} \right)} & \text{otherwise} \end{cases} 
Piecewise(((37*cos(407/50)^2/20 + 37*sin(407/50)^2/20 + 5*cos(407/50)*sin(407/50)/22)*cos(23*u/10) - (23*cos(253/50)^2/20 + 23*sin(253/50)^2/20 + 5*cos(253/50)*sin(253/50)/22)*cos(23*u/10) - 5*sin(407/50)^2*sin(23*u/10)/22 + 5*sin(253/50)^2*sin(23*u/10)/22, u = -11/5), ((37*cos(407/50)^2/20 + 37*sin(407/50)^2/20 + 5*cos(407/50)*sin(407/50)/22)*cos(23*u/10) - (23*cos(253/50)^2/20 + 23*sin(253/50)^2/20 + 5*cos(253/50)*sin(253/50)/22)*cos(23*u/10) - 5*sin(253/50)^2*sin(23*u/10)/22 + 5*sin(407/50)^2*sin(23*u/10)/22, u = 11/5), ((-55*cos(37*u/10)*sin(407/50)/(-121 + 25*u^2) + 25*u*cos(407/50)*sin(37*u/10)/(-121 + 25*u^2))*cos(23*u/10) + (-55*sin(407/50)*sin(37*u/10)/(-121 + 25*u^2) - 25*u*cos(407/50)*cos(37*u/10)/(-121 + 25*u^2))*sin(23*u/10) - (-55*cos(23*u/10)*sin(253/50)/(-121 + 25*u^2) + 25*u*cos(253/50)*sin(23*u/10)/(-121 + 25*u^2))*cos(23*u/10) - (-55*sin(253/50)*sin(23*u/10)/(-121 + 25*u^2) - 25*u*cos(253/50)*cos(23*u/10)/(-121 + 25*u^2))*sin(23*u/10), True))

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

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