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Resolución de integrales/ sin(ax)/ cos(ax)/ (cos(ax)+sin(ax))^2

Integral de (cos(ax)+sin(ax))^2 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

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

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

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