Sr Examen
Integral de cos(nx)cos(mx) dx
Solución
Ha introducido
[src]
pi / | | cos(n*x)*cos(m*x) dx | / -pi
$$\int\limits_{- \pi}^{\pi} \cos{\left(m x \right)} \cos{\left(n x \right)}\, dx$$
Integral(cos(n*x)*cos(m*x), (x, -pi, pi))
Respuesta (Indefinida)
[src]
// x for And(m = 0, n = 0)\
|| |
|| 2 2 |
/ ||x*cos (n*x) x*sin (n*x) cos(n*x)*sin(n*x) |
| ||----------- + ----------- + ----------------- for Or(m = -n, m = n)|
| cos(n*x)*cos(m*x) dx = C + |< 2 2 2*n |
| || |
/ || m*cos(n*x)*sin(m*x) n*cos(m*x)*sin(n*x) |
|| ------------------- - ------------------- otherwise |
|| 2 2 2 2 |
\\ m - n m - n /
$$\int \cos{\left(m x \right)} \cos{\left(n x \right)}\, dx = C + \begin{cases} x & \text{for}\: m = 0 \wedge n = 0 \\\frac{x \sin^{2}{\left(n x \right)}}{2} + \frac{x \cos^{2}{\left(n x \right)}}{2} + \frac{\sin{\left(n x \right)} \cos{\left(n x \right)}}{2 n} & \text{for}\: m = - n \vee m = n \\\frac{m \sin{\left(m x \right)} \cos{\left(n x \right)}}{m^{2} - n^{2}} - \frac{n \sin{\left(n x \right)} \cos{\left(m x \right)}}{m^{2} - n^{2}} & \text{otherwise} \end{cases}$$
Respuesta
[src]
/ 2*pi for Or(And(m = 0, n = 0), And(m = 0, m = n, n = 0), And(m = 0, m = -n, n = 0), And(m = 0, m = -n, m = n, n = 0)) | | 2 2 cos(pi*n)*sin(pi*n) |pi*cos (pi*n) + pi*sin (pi*n) + ------------------- for Or(And(m = 0, m = -n), And(m = 0, m = n), And(m = -n, m = n), And(m = n, n = 0), And(m = -n, n = 0), And(m = 0, m = -n, m = n), And(m = -n, m = n, n = 0), m = -n, m = n) | n < | 2*n*cos(pi*m)*sin(pi*n) 2*m*cos(pi*n)*sin(pi*m) |- ----------------------- + ----------------------- otherwise | 2 2 2 2 | m - n m - n \
$$\begin{cases} 2 \pi & \text{for}\: \left(m = 0 \wedge n = 0\right) \vee \left(m = 0 \wedge m = n \wedge n = 0\right) \vee \left(m = 0 \wedge m = - n \wedge n = 0\right) \vee \left(m = 0 \wedge m = - n \wedge m = n \wedge n = 0\right) \\\pi \sin^{2}{\left(\pi n \right)} + \pi \cos^{2}{\left(\pi n \right)} + \frac{\sin{\left(\pi n \right)} \cos{\left(\pi n \right)}}{n} & \text{for}\: \left(m = 0 \wedge m = - n\right) \vee \left(m = 0 \wedge m = n\right) \vee \left(m = - n \wedge m = n\right) \vee \left(m = n \wedge n = 0\right) \vee \left(m = - n \wedge n = 0\right) \vee \left(m = 0 \wedge m = - n \wedge m = n\right) \vee \left(m = - n \wedge m = n \wedge n = 0\right) \vee m = - n \vee m = n \\\frac{2 m \sin{\left(\pi m \right)} \cos{\left(\pi n \right)}}{m^{2} - n^{2}} - \frac{2 n \sin{\left(\pi n \right)} \cos{\left(\pi m \right)}}{m^{2} - n^{2}} & \text{otherwise} \end{cases}$$
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/ 2*pi for Or(And(m = 0, n = 0), And(m = 0, m = n, n = 0), And(m = 0, m = -n, n = 0), And(m = 0, m = -n, m = n, n = 0)) | | 2 2 cos(pi*n)*sin(pi*n) |pi*cos (pi*n) + pi*sin (pi*n) + ------------------- for Or(And(m = 0, m = -n), And(m = 0, m = n), And(m = -n, m = n), And(m = n, n = 0), And(m = -n, n = 0), And(m = 0, m = -n, m = n), And(m = -n, m = n, n = 0), m = -n, m = n) | n < | 2*n*cos(pi*m)*sin(pi*n) 2*m*cos(pi*n)*sin(pi*m) |- ----------------------- + ----------------------- otherwise | 2 2 2 2 | m - n m - n \
$$\begin{cases} 2 \pi & \text{for}\: \left(m = 0 \wedge n = 0\right) \vee \left(m = 0 \wedge m = n \wedge n = 0\right) \vee \left(m = 0 \wedge m = - n \wedge n = 0\right) \vee \left(m = 0 \wedge m = - n \wedge m = n \wedge n = 0\right) \\\pi \sin^{2}{\left(\pi n \right)} + \pi \cos^{2}{\left(\pi n \right)} + \frac{\sin{\left(\pi n \right)} \cos{\left(\pi n \right)}}{n} & \text{for}\: \left(m = 0 \wedge m = - n\right) \vee \left(m = 0 \wedge m = n\right) \vee \left(m = - n \wedge m = n\right) \vee \left(m = n \wedge n = 0\right) \vee \left(m = - n \wedge n = 0\right) \vee \left(m = 0 \wedge m = - n \wedge m = n\right) \vee \left(m = - n \wedge m = n \wedge n = 0\right) \vee m = - n \vee m = n \\\frac{2 m \sin{\left(\pi m \right)} \cos{\left(\pi n \right)}}{m^{2} - n^{2}} - \frac{2 n \sin{\left(\pi n \right)} \cos{\left(\pi m \right)}}{m^{2} - n^{2}} & \text{otherwise} \end{cases}$$
Piecewise((2*pi, ((m = 0)∧(n = 0))∨((m = 0)∧(m = n)∧(n = 0))∨((m = 0)∧(n = 0)∧(m = -n))∨((m = 0)∧(m = n)∧(n = 0)∧(m = -n))), (pi*cos(pi*n)^2 + pi*sin(pi*n)^2 + cos(pi*n)*sin(pi*n)/n, (m = n)∨(m = -n)∨((m = 0)∧(m = n))∨((m = n)∧(n = 0))∨((m = 0)∧(m = -n))∨((m = n)∧(m = -n))∨((n = 0)∧(m = -n))∨((m = 0)∧(m = n)∧(m = -n))∨((m = n)∧(n = 0)∧(m = -n))), (-2*n*cos(pi*m)*sin(pi*n)/(m^2 - n^2) + 2*m*cos(pi*n)*sin(pi*m)/(m^2 - n^2), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.