Integral de cos(x)×cos(ax) dx
Solución
Respuesta (Indefinida)
[src]
// 2 2 \
||x*cos (x) x*sin (x) cos(x)*sin(x) |
/ ||--------- + --------- + ------------- for Or(a = -1, a = 1)|
| || 2 2 2 |
| cos(x)*cos(a*x) dx = C + |< |
| || cos(a*x)*sin(x) a*cos(x)*sin(a*x) |
/ ||- --------------- + ----------------- otherwise |
|| 2 2 |
\\ -1 + a -1 + a /
$$\int \cos{\left(x \right)} \cos{\left(a x \right)}\, dx = C + \begin{cases} \frac{x \sin^{2}{\left(x \right)}}{2} + \frac{x \cos^{2}{\left(x \right)}}{2} + \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{2} & \text{for}\: a = -1 \vee a = 1 \\\frac{a \sin{\left(a x \right)} \cos{\left(x \right)}}{a^{2} - 1} - \frac{\sin{\left(x \right)} \cos{\left(a x \right)}}{a^{2} - 1} & \text{otherwise} \end{cases}$$
/ 2 2
|cos (1) sin (1) cos(1)*sin(1)
|------- + ------- + ------------- for Or(a = -1, a = 1)
| 2 2 2
<
| cos(a)*sin(1) a*cos(1)*sin(a)
|- ------------- + --------------- otherwise
| 2 2
\ -1 + a -1 + a
$$\begin{cases} \frac{\cos^{2}{\left(1 \right)}}{2} + \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{\sin^{2}{\left(1 \right)}}{2} & \text{for}\: a = -1 \vee a = 1 \\\frac{a \sin{\left(a \right)} \cos{\left(1 \right)}}{a^{2} - 1} - \frac{\sin{\left(1 \right)} \cos{\left(a \right)}}{a^{2} - 1} & \text{otherwise} \end{cases}$$
=
/ 2 2
|cos (1) sin (1) cos(1)*sin(1)
|------- + ------- + ------------- for Or(a = -1, a = 1)
| 2 2 2
<
| cos(a)*sin(1) a*cos(1)*sin(a)
|- ------------- + --------------- otherwise
| 2 2
\ -1 + a -1 + a
$$\begin{cases} \frac{\cos^{2}{\left(1 \right)}}{2} + \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{\sin^{2}{\left(1 \right)}}{2} & \text{for}\: a = -1 \vee a = 1 \\\frac{a \sin{\left(a \right)} \cos{\left(1 \right)}}{a^{2} - 1} - \frac{\sin{\left(1 \right)} \cos{\left(a \right)}}{a^{2} - 1} & \text{otherwise} \end{cases}$$
Piecewise((cos(1)^2/2 + sin(1)^2/2 + cos(1)*sin(1)/2, (a = -1)∨(a = 1)), (-cos(a)*sin(1)/(-1 + a^2) + a*cos(1)*sin(a)/(-1 + a^2), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.