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Resolución de integrales/ e^(a*x)/ (e^(a*x)-sin(b*x))^2

Integral de (e^(a*x)-sin(b*x))^2 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
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∫01(eax−sin⁡(bx))2 dx\int\limits_{0}^{1} \left(e^{a x} - \sin{\left(b x \right)}\right)^{2}\, dx 
Integral((E^(a*x) - sin(b*x))^2, (x, 0, 1))
Respuesta (Indefinida) [src] 
                                                                                                //                         /cos(b*x)   x*sin(b*x)                                                \                       
                                                                                                ||                         |-------- + ----------  for b != 0                                    |                       
                                                                                                ||                         |    2          b                                                     |                       
                                                                                                ||                         |   b                                                                 |                       
                                                                                                ||                         <                                                            for a = 0|                       
                                                                                                ||                         |          2                                                          |                       
                                                                                                ||                         |         x                                                           |                       
                                                                                                ||                         |         --            otherwise                                     |                       
                                                                                                ||                         \         2                                                           |                       
                                   /    x       for b = 0                                       ||                                                                                               |                       
  /                                |                                                            ||/                             x                                for And(a = 0, b = 0)           |                       
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∫(eax−sin⁡(bx))2 dx=C+2b({{xsin⁡(bx)b+cos⁡(bx)b2for b≠0x22otherwisefor a=0{xfor a=0∧b=0ixe−ibxsin⁡(bx)2+xe−ibxcos⁡(bx)2+e−ibxsin⁡(bx)2bfor a=−ib−ixeibxsin⁡(bx)2+xeibxcos⁡(bx)2+eibxsin⁡(bx)2bfor a=ibaeaxcos⁡(bx)a2+b2+beaxsin⁡(bx)a2+b2otherwiseaotherwise)+x2−2({xfor a=0eaxaotherwise)sin⁡(bx)+{xfor a=0e2ax2aotherwise−{xfor b=0sin⁡(2bx)2botherwise2\int \left(e^{a x} - \sin{\left(b x \right)}\right)^{2}\, dx = C + 2 b \left(\begin{cases} \begin{cases} \frac{x \sin{\left(b x \right)}}{b} + \frac{\cos{\left(b x \right)}}{b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases} & \text{for}\: a = 0 \\\frac{\begin{cases} x & \text{for}\: a = 0 \wedge b = 0 \\\frac{i x e^{- i b x} \sin{\left(b x \right)}}{2} + \frac{x e^{- i b x} \cos{\left(b x \right)}}{2} + \frac{e^{- i b x} \sin{\left(b x \right)}}{2 b} & \text{for}\: a = - i b \\- \frac{i x e^{i b x} \sin{\left(b x \right)}}{2} + \frac{x e^{i b x} \cos{\left(b x \right)}}{2} + \frac{e^{i b x} \sin{\left(b x \right)}}{2 b} & \text{for}\: a = i b \\\frac{a e^{a x} \cos{\left(b x \right)}}{a^{2} + b^{2}} + \frac{b e^{a x} \sin{\left(b x \right)}}{a^{2} + b^{2}} & \text{otherwise} \end{cases}}{a} & \text{otherwise} \end{cases}\right) + \frac{x}{2} - 2 \left(\begin{cases} x & \text{for}\: a = 0 \\\frac{e^{a x}}{a} & \text{otherwise} \end{cases}\right) \sin{\left(b x \right)} + \begin{cases} x & \text{for}\: a = 0 \\\frac{e^{2 a x}}{2 a} & \text{otherwise} \end{cases} - \frac{\begin{cases} x & \text{for}\: b = 0 \\\frac{\sin{\left(2 b x \right)}}{2 b} & \text{otherwise} \end{cases}}{2}
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Respuesta [src] 
/                                                                                                                       1                                                                                                                         for Or(And(a = 0, b = 0), And(a = 0, a = -I*b, b = 0), And(a = 0, a = I*b, b = 0), And(a = 0, a = -I*b, a = I*b, b = 0))
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|                                                                                                    2         2                                                                                                                                                                                                                                                          
|                                                                                                 cos (b)   sin (b)   2   2*cos(b)   cos(b)*sin(b)                                                                                                                                                                                                                        
|                                                                                             1 + ------- + ------- - - + -------- - -------------                                                                                                                for Or(And(a = 0, a = -I*b), And(a = 0, a = I*b), And(a = 0, a = -I*b, a = I*b), a = 0)                 
|                                                                                                    2         2      b      b            2*b                                                                                                                                                                                                                             
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|                                                                        2         2                                               -2*I*b                      -I*b                                                                                                                                                                                                       
|                                                                     cos (b)   sin (b)    -I*b           I              -I*b   I*e         cos(b)*sin(b)   I*e    *sin(b)                                                                                                                                                                                                
|                                                                     ------- + ------- - e    *sin(b) - --- + I*cos(b)*e     + --------- - ------------- - --------------                                                                                     for Or(And(a = -I*b, a = I*b), And(a = -I*b, b = 0), And(a = -I*b, a = I*b, b = 0), a = -I*b)              
|                                                                        2         2                     2*b                       2*b           2*b              b                                                                                                                                                                                                       
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|                                                                          2         2                                             2*I*b                      I*b                                                                                                                                                                                                         
<                                                                       cos (b)   sin (b)    I     I*b                    I*b   I*e        cos(b)*sin(b)   I*e   *sin(b)                                                                                                                                                                                                  
|                                                                       ------- + ------- + --- - e   *sin(b) - I*cos(b)*e    - -------- - ------------- + -------------                                                                                                                    for Or(And(a = I*b, b = 0), a = I*b)                                          
|                                                                          2         2      2*b                                   2*b           2*b              b                                                                                                                                                                                                        
|                                                                                                                                                                                                                                                                                                                                                                         
|                                                                                                                          2*a                                                                                                                                                                                                                                            
|                                                                                                                    1    e                                                                                                                                                                                                                                               
|                                                                                                                 - --- + ----                                                                                                                                                                           for b = 0                                                        
|                                                                                                                   2*a   2*a                                                                                                                                                                                                                                             
|                                                                                                                                                                                                                                                                                                                                                                         
|          3              3  2*a                2                  2            3    2            3    2             2  2*a           3    2            3    2        3                    2                      2  a               2         a                                                                                                                          
|         b              b *e                b*a              4*a*b          a*b *cos (b)      a*b *sin (b)       b*a *e           b*a *cos (b)      b*a *sin (b)    a *cos(b)*sin(b)   a*b *cos(b)*sin(b)   4*b*a *e *sin(b)   4*a*b *cos(b)*e                                                                                                                           
|- --------------- + --------------- - --------------- - --------------- + --------------- + --------------- + --------------- + --------------- + --------------- - ---------------- - ------------------ - ---------------- + ----------------                                                         otherwise                                                        
|       3        3        3        3        3        3        3        3        3        3        3        3        3        3        3        3        3        3        3        3          3        3          3        3         3        3                                                                                                                           
\  2*a*b  + 2*b*a    2*a*b  + 2*b*a    2*a*b  + 2*b*a    2*a*b  + 2*b*a    2*a*b  + 2*b*a    2*a*b  + 2*b*a    2*a*b  + 2*b*a    2*a*b  + 2*b*a    2*a*b  + 2*b*a    2*a*b  + 2*b*a      2*a*b  + 2*b*a      2*a*b  + 2*b*a     2*a*b  + 2*b*a
{1for (a=0∧b=0)∨(a=0∧a=−ib∧b=0)∨(a=0∧a=ib∧b=0)∨(a=0∧a=−ib∧a=ib∧b=0)sin⁡2(b)2+cos⁡2(b)2+1−sin⁡(b)cos⁡(b)2b+2cos⁡(b)b−2bfor (a=0∧a=−ib)∨(a=0∧a=ib)∨(a=0∧a=−ib∧a=ib)∨a=0sin⁡2(b)2+cos⁡2(b)2−e−ibsin⁡(b)+ie−ibcos⁡(b)−sin⁡(b)cos⁡(b)2b−i2b−ie−ibsin⁡(b)b+ie−2ib2bfor (a=−ib∧a=ib)∨(a=−ib∧b=0)∨(a=−ib∧a=ib∧b=0)∨a=−ib−eibsin⁡(b)−ieibcos⁡(b)+sin⁡2(b)2+cos⁡2(b)2−ie2ib2b+ieibsin⁡(b)b−sin⁡(b)cos⁡(b)2b+i2bfor (a=ib∧b=0)∨a=ibe2a2a−12afor b=0a3bsin⁡2(b)2a3b+2ab3+a3bcos⁡2(b)2a3b+2ab3−a3sin⁡(b)cos⁡(b)2a3b+2ab3+a2be2a2a3b+2ab3−4a2beasin⁡(b)2a3b+2ab3−a2b2a3b+2ab3+ab3sin⁡2(b)2a3b+2ab3+ab3cos⁡2(b)2a3b+2ab3+4ab2eacos⁡(b)2a3b+2ab3−ab2sin⁡(b)cos⁡(b)2a3b+2ab3−4ab22a3b+2ab3+b3e2a2a3b+2ab3−b32a3b+2ab3otherwise\begin{cases} 1 & \text{for}\: \left(a = 0 \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge a = i b \wedge b = 0\right) \\\frac{\sin^{2}{\left(b \right)}}{2} + \frac{\cos^{2}{\left(b \right)}}{2} + 1 - \frac{\sin{\left(b \right)} \cos{\left(b \right)}}{2 b} + \frac{2 \cos{\left(b \right)}}{b} - \frac{2}{b} & \text{for}\: \left(a = 0 \wedge a = - i b\right) \vee \left(a = 0 \wedge a = i b\right) \vee \left(a = 0 \wedge a = - i b \wedge a = i b\right) \vee a = 0 \\\frac{\sin^{2}{\left(b \right)}}{2} + \frac{\cos^{2}{\left(b \right)}}{2} - e^{- i b} \sin{\left(b \right)} + i e^{- i b} \cos{\left(b \right)} - \frac{\sin{\left(b \right)} \cos{\left(b \right)}}{2 b} - \frac{i}{2 b} - \frac{i e^{- i b} \sin{\left(b \right)}}{b} + \frac{i e^{- 2 i b}}{2 b} & \text{for}\: \left(a = - i b \wedge a = i b\right) \vee \left(a = - i b \wedge b = 0\right) \vee \left(a = - i b \wedge a = i b \wedge b = 0\right) \vee a = - i b \\- e^{i b} \sin{\left(b \right)} - i e^{i b} \cos{\left(b \right)} + \frac{\sin^{2}{\left(b \right)}}{2} + \frac{\cos^{2}{\left(b \right)}}{2} - \frac{i e^{2 i b}}{2 b} + \frac{i e^{i b} \sin{\left(b \right)}}{b} - \frac{\sin{\left(b \right)} \cos{\left(b \right)}}{2 b} + \frac{i}{2 b} & \text{for}\: \left(a = i b \wedge b = 0\right) \vee a = i b \\\frac{e^{2 a}}{2 a} - \frac{1}{2 a} & \text{for}\: b = 0 \\\frac{a^{3} b \sin^{2}{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} + \frac{a^{3} b \cos^{2}{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} - \frac{a^{3} \sin{\left(b \right)} \cos{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} + \frac{a^{2} b e^{2 a}}{2 a^{3} b + 2 a b^{3}} - \frac{4 a^{2} b e^{a} \sin{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} - \frac{a^{2} b}{2 a^{3} b + 2 a b^{3}} + \frac{a b^{3} \sin^{2}{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} + \frac{a b^{3} \cos^{2}{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} + \frac{4 a b^{2} e^{a} \cos{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} - \frac{a b^{2} \sin{\left(b \right)} \cos{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} - \frac{4 a b^{2}}{2 a^{3} b + 2 a b^{3}} + \frac{b^{3} e^{2 a}}{2 a^{3} b + 2 a b^{3}} - \frac{b^{3}}{2 a^{3} b + 2 a b^{3}} & \text{otherwise} \end{cases} 
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/                                                                                                                       1                                                                                                                         for Or(And(a = 0, b = 0), And(a = 0, a = -I*b, b = 0), And(a = 0, a = I*b, b = 0), And(a = 0, a = -I*b, a = I*b, b = 0))
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|                                                                                                    2         2                                                                                                                                                                                                                                                          
|                                                                                                 cos (b)   sin (b)   2   2*cos(b)   cos(b)*sin(b)                                                                                                                                                                                                                        
|                                                                                             1 + ------- + ------- - - + -------- - -------------                                                                                                                for Or(And(a = 0, a = -I*b), And(a = 0, a = I*b), And(a = 0, a = -I*b, a = I*b), a = 0)                 
|                                                                                                    2         2      b      b            2*b                                                                                                                                                                                                                             
|                                                                                                                                                                                                                                                                                                                                                                         
|                                                                        2         2                                               -2*I*b                      -I*b                                                                                                                                                                                                       
|                                                                     cos (b)   sin (b)    -I*b           I              -I*b   I*e         cos(b)*sin(b)   I*e    *sin(b)                                                                                                                                                                                                
|                                                                     ------- + ------- - e    *sin(b) - --- + I*cos(b)*e     + --------- - ------------- - --------------                                                                                     for Or(And(a = -I*b, a = I*b), And(a = -I*b, b = 0), And(a = -I*b, a = I*b, b = 0), a = -I*b)              
|                                                                        2         2                     2*b                       2*b           2*b              b                                                                                                                                                                                                       
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|                                                                          2         2                                             2*I*b                      I*b                                                                                                                                                                                                         
<                                                                       cos (b)   sin (b)    I     I*b                    I*b   I*e        cos(b)*sin(b)   I*e   *sin(b)                                                                                                                                                                                                  
|                                                                       ------- + ------- + --- - e   *sin(b) - I*cos(b)*e    - -------- - ------------- + -------------                                                                                                                    for Or(And(a = I*b, b = 0), a = I*b)                                          
|                                                                          2         2      2*b                                   2*b           2*b              b                                                                                                                                                                                                        
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|                                                                                                                          2*a                                                                                                                                                                                                                                            
|                                                                                                                    1    e                                                                                                                                                                                                                                               
|                                                                                                                 - --- + ----                                                                                                                                                                           for b = 0                                                        
|                                                                                                                   2*a   2*a                                                                                                                                                                                                                                             
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|          3              3  2*a                2                  2            3    2            3    2             2  2*a           3    2            3    2        3                    2                      2  a               2         a                                                                                                                          
|         b              b *e                b*a              4*a*b          a*b *cos (b)      a*b *sin (b)       b*a *e           b*a *cos (b)      b*a *sin (b)    a *cos(b)*sin(b)   a*b *cos(b)*sin(b)   4*b*a *e *sin(b)   4*a*b *cos(b)*e                                                                                                                           
|- --------------- + --------------- - --------------- - --------------- + --------------- + --------------- + --------------- + --------------- + --------------- - ---------------- - ------------------ - ---------------- + ----------------                                                         otherwise                                                        
|       3        3        3        3        3        3        3        3        3        3        3        3        3        3        3        3        3        3        3        3          3        3          3        3         3        3                                                                                                                           
\  2*a*b  + 2*b*a    2*a*b  + 2*b*a    2*a*b  + 2*b*a    2*a*b  + 2*b*a    2*a*b  + 2*b*a    2*a*b  + 2*b*a    2*a*b  + 2*b*a    2*a*b  + 2*b*a    2*a*b  + 2*b*a    2*a*b  + 2*b*a      2*a*b  + 2*b*a      2*a*b  + 2*b*a     2*a*b  + 2*b*a
{1for (a=0∧b=0)∨(a=0∧a=−ib∧b=0)∨(a=0∧a=ib∧b=0)∨(a=0∧a=−ib∧a=ib∧b=0)sin⁡2(b)2+cos⁡2(b)2+1−sin⁡(b)cos⁡(b)2b+2cos⁡(b)b−2bfor (a=0∧a=−ib)∨(a=0∧a=ib)∨(a=0∧a=−ib∧a=ib)∨a=0sin⁡2(b)2+cos⁡2(b)2−e−ibsin⁡(b)+ie−ibcos⁡(b)−sin⁡(b)cos⁡(b)2b−i2b−ie−ibsin⁡(b)b+ie−2ib2bfor (a=−ib∧a=ib)∨(a=−ib∧b=0)∨(a=−ib∧a=ib∧b=0)∨a=−ib−eibsin⁡(b)−ieibcos⁡(b)+sin⁡2(b)2+cos⁡2(b)2−ie2ib2b+ieibsin⁡(b)b−sin⁡(b)cos⁡(b)2b+i2bfor (a=ib∧b=0)∨a=ibe2a2a−12afor b=0a3bsin⁡2(b)2a3b+2ab3+a3bcos⁡2(b)2a3b+2ab3−a3sin⁡(b)cos⁡(b)2a3b+2ab3+a2be2a2a3b+2ab3−4a2beasin⁡(b)2a3b+2ab3−a2b2a3b+2ab3+ab3sin⁡2(b)2a3b+2ab3+ab3cos⁡2(b)2a3b+2ab3+4ab2eacos⁡(b)2a3b+2ab3−ab2sin⁡(b)cos⁡(b)2a3b+2ab3−4ab22a3b+2ab3+b3e2a2a3b+2ab3−b32a3b+2ab3otherwise\begin{cases} 1 & \text{for}\: \left(a = 0 \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge a = i b \wedge b = 0\right) \\\frac{\sin^{2}{\left(b \right)}}{2} + \frac{\cos^{2}{\left(b \right)}}{2} + 1 - \frac{\sin{\left(b \right)} \cos{\left(b \right)}}{2 b} + \frac{2 \cos{\left(b \right)}}{b} - \frac{2}{b} & \text{for}\: \left(a = 0 \wedge a = - i b\right) \vee \left(a = 0 \wedge a = i b\right) \vee \left(a = 0 \wedge a = - i b \wedge a = i b\right) \vee a = 0 \\\frac{\sin^{2}{\left(b \right)}}{2} + \frac{\cos^{2}{\left(b \right)}}{2} - e^{- i b} \sin{\left(b \right)} + i e^{- i b} \cos{\left(b \right)} - \frac{\sin{\left(b \right)} \cos{\left(b \right)}}{2 b} - \frac{i}{2 b} - \frac{i e^{- i b} \sin{\left(b \right)}}{b} + \frac{i e^{- 2 i b}}{2 b} & \text{for}\: \left(a = - i b \wedge a = i b\right) \vee \left(a = - i b \wedge b = 0\right) \vee \left(a = - i b \wedge a = i b \wedge b = 0\right) \vee a = - i b \\- e^{i b} \sin{\left(b \right)} - i e^{i b} \cos{\left(b \right)} + \frac{\sin^{2}{\left(b \right)}}{2} + \frac{\cos^{2}{\left(b \right)}}{2} - \frac{i e^{2 i b}}{2 b} + \frac{i e^{i b} \sin{\left(b \right)}}{b} - \frac{\sin{\left(b \right)} \cos{\left(b \right)}}{2 b} + \frac{i}{2 b} & \text{for}\: \left(a = i b \wedge b = 0\right) \vee a = i b \\\frac{e^{2 a}}{2 a} - \frac{1}{2 a} & \text{for}\: b = 0 \\\frac{a^{3} b \sin^{2}{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} + \frac{a^{3} b \cos^{2}{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} - \frac{a^{3} \sin{\left(b \right)} \cos{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} + \frac{a^{2} b e^{2 a}}{2 a^{3} b + 2 a b^{3}} - \frac{4 a^{2} b e^{a} \sin{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} - \frac{a^{2} b}{2 a^{3} b + 2 a b^{3}} + \frac{a b^{3} \sin^{2}{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} + \frac{a b^{3} \cos^{2}{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} + \frac{4 a b^{2} e^{a} \cos{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} - \frac{a b^{2} \sin{\left(b \right)} \cos{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} - \frac{4 a b^{2}}{2 a^{3} b + 2 a b^{3}} + \frac{b^{3} e^{2 a}}{2 a^{3} b + 2 a b^{3}} - \frac{b^{3}}{2 a^{3} b + 2 a b^{3}} & \text{otherwise} \end{cases} 
Piecewise((1, ((a = 0)∧(b = 0))∨((a = 0)∧(b = 0)∧(a = i*b))∨((a = 0)∧(b = 0)∧(a = -i*b))∨((a = 0)∧(b = 0)∧(a = i*b)∧(a = -i*b))), (1 + cos(b)^2/2 + sin(b)^2/2 - 2/b + 2*cos(b)/b - cos(b)*sin(b)/(2*b), (a = 0)∨((a = 0)∧(a = i*b))∨((a = 0)∧(a = -i*b))∨((a = 0)∧(a = i*b)∧(a = -i*b))), (cos(b)^2/2 + sin(b)^2/2 - exp(-i*b)*sin(b) - i/(2*b) + i*cos(b)*exp(-i*b) + i*exp(-2*i*b)/(2*b) - cos(b)*sin(b)/(2*b) - i*exp(-i*b)*sin(b)/b, (a = -i*b)∨((b = 0)∧(a = -i*b))∨((a = i*b)∧(a = -i*b))∨((b = 0)∧(a = i*b)∧(a = -i*b))), (cos(b)^2/2 + sin(b)^2/2 + i/(2*b) - exp(i*b)*sin(b) - i*cos(b)*exp(i*b) - i*exp(2*i*b)/(2*b) - cos(b)*sin(b)/(2*b) + i*exp(i*b)*sin(b)/b, (a = i*b)∨((b = 0)∧(a = i*b))), (-1/(2*a) + exp(2*a)/(2*a), b = 0), (-b^3/(2*a*b^3 + 2*b*a^3) + b^3*exp(2*a)/(2*a*b^3 + 2*b*a^3) - b*a^2/(2*a*b^3 + 2*b*a^3) - 4*a*b^2/(2*a*b^3 + 2*b*a^3) + a*b^3*cos(b)^2/(2*a*b^3 + 2*b*a^3) + a*b^3*sin(b)^2/(2*a*b^3 + 2*b*a^3) + b*a^2*exp(2*a)/(2*a*b^3 + 2*b*a^3) + b*a^3*cos(b)^2/(2*a*b^3 + 2*b*a^3) + b*a^3*sin(b)^2/(2*a*b^3 + 2*b*a^3) - a^3*cos(b)*sin(b)/(2*a*b^3 + 2*b*a^3) - a*b^2*cos(b)*sin(b)/(2*a*b^3 + 2*b*a^3) - 4*b*a^2*exp(a)*sin(b)/(2*a*b^3 + 2*b*a^3) + 4*a*b^2*cos(b)*exp(a)/(2*a*b^3 + 2*b*a^3), True))

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

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