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Resolución de integrales/ sin(nx)/ sin(nx)-(n^x)+((x-n)/x^2)

Integral de sin(nx)-(n^x)+((x-n)/x^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                           
  /                           
 |                            
 |  /            x   x - n\   
 |  |sin(n*x) - n  + -----| dx
 |  |                   2 |   
 |  \                  x  /   
 |                            
/                             
0
$$\int\limits_{0}^{1} \left(\left(- n^{x} + \sin{\left(n x \right)}\right) + \frac{- n + x}{x^{2}}\right)\, dx$$ 
Integral(sin(n*x) - n^x + (x - n)/x^2, (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                 //   x                   \                                        
 |                                  ||  n                    |       //    0       for n = 0\         
 | /            x   x - n\          ||------  for log(n) != 0|   n   ||                     |         
 | |sin(n*x) - n  + -----| dx = C - |
$$\int \left(\left(- n^{x} + \sin{\left(n x \right)}\right) + \frac{- n + x}{x^{2}}\right)\, dx = C + \frac{n}{x} + \begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\cos{\left(n x \right)}}{n} & \text{otherwise} \end{cases} - \begin{cases} \frac{n^{x}}{\log{\left(n \right)}} & \text{for}\: \log{\left(n \right)} \neq 0 \\x & \text{otherwise} \end{cases} + \log{\left(x \right)}$$
Simplificar
Respuesta [src] 
                 //1     1        n      cos(n)                                                 \
                 ||- + ------ - ------ - ------   for Or(And(n > 0, n < 1), And(n > 1, n < oo)) |
                 ||n   log(n)   log(n)     n                                                    |
                 ||                                                                             |
                 ||           1   cos(n)                                                        |
                 ||      -1 + - - ------                for And(n > -oo, n < oo, n != 0)        |
n - oo*sign(n) + |<           n     n                                                           |
                 ||                                                                             |
                 ||        1        n                                                           |
                 ||      ------ - ------         for Or(And(n >= 0, n < 1), And(n <= oo, n > 1))|
                 ||      log(n)   log(n)                                                        |
                 ||                                                                             |
                 \\             -1                                  otherwise                   /
$$n + \begin{cases} - \frac{n}{\log{\left(n \right)}} + \frac{1}{\log{\left(n \right)}} - \frac{\cos{\left(n \right)}}{n} + \frac{1}{n} & \text{for}\: \left(n > 0 \wedge n < 1\right) \vee \left(n > 1 \wedge n < \infty\right) \\-1 - \frac{\cos{\left(n \right)}}{n} + \frac{1}{n} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\- \frac{n}{\log{\left(n \right)}} + \frac{1}{\log{\left(n \right)}} & \text{for}\: \left(n \geq 0 \wedge n < 1\right) \vee \left(n \leq \infty \wedge n > 1\right) \\-1 & \text{otherwise} \end{cases} - \infty \operatorname{sign}{\left(n \right)}$$ 
=
= 
                 //1     1        n      cos(n)                                                 \
                 ||- + ------ - ------ - ------   for Or(And(n > 0, n < 1), And(n > 1, n < oo)) |
                 ||n   log(n)   log(n)     n                                                    |
                 ||                                                                             |
                 ||           1   cos(n)                                                        |
                 ||      -1 + - - ------                for And(n > -oo, n < oo, n != 0)        |
n - oo*sign(n) + |<           n     n                                                           |
                 ||                                                                             |
                 ||        1        n                                                           |
                 ||      ------ - ------         for Or(And(n >= 0, n < 1), And(n <= oo, n > 1))|
                 ||      log(n)   log(n)                                                        |
                 ||                                                                             |
                 \\             -1                                  otherwise                   /
$$n + \begin{cases} - \frac{n}{\log{\left(n \right)}} + \frac{1}{\log{\left(n \right)}} - \frac{\cos{\left(n \right)}}{n} + \frac{1}{n} & \text{for}\: \left(n > 0 \wedge n < 1\right) \vee \left(n > 1 \wedge n < \infty\right) \\-1 - \frac{\cos{\left(n \right)}}{n} + \frac{1}{n} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\- \frac{n}{\log{\left(n \right)}} + \frac{1}{\log{\left(n \right)}} & \text{for}\: \left(n \geq 0 \wedge n < 1\right) \vee \left(n \leq \infty \wedge n > 1\right) \\-1 & \text{otherwise} \end{cases} - \infty \operatorname{sign}{\left(n \right)}$$ 
n - oo*sign(n) + Piecewise((1/n + 1/log(n) - n/log(n) - cos(n)/n, ((n > 0)∧(n < 1))∨((n > 1)∧(n < oo))), (-1 + 1/n - cos(n)/n, (n > -oo)∧(n < oo)∧(Ne(n, 0))), (1/log(n) - n/log(n), ((n >= 0)∧(n < 1))∨((n <= oo)∧(n > 1))), (-1, True))

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

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