Sr Examen
Integral de -cos(2*x)*cos(n*x) dx
Solución
Ha introducido
[src]
3*pi ---- 4 / | | -cos(2*x)*cos(n*x) dx | / pi -- 4
$$\int\limits_{\frac{\pi}{4}}^{\frac{3 \pi}{4}} - \cos{\left(2 x \right)} \cos{\left(n x \right)}\, dx$$
Integral((-cos(2*x))*cos(n*x), (x, pi/4, 3*pi/4))
Respuesta (Indefinida)
[src]
// 2 2 2 \
||sin (x)*sin(2*x) x*sin (x)*cos(2*x) x*cos (x)*cos(2*x) 3*cos(x)*cos(2*x)*sin(x) x*cos(x)*sin(x)*sin(2*x) |
||---------------- - ------------------ + ------------------ + ------------------------ + ------------------------ for n = -2|
|| 2 4 4 4 2 |
|| |
|| 2 2 |
|| x*cos (x) x*sin (x) cos(x)*sin(x) |
|| --------- + --------- + ------------- for n = 0 |
/ || 2 2 2 | // x for n = 0\
| || | || |
| -cos(2*x)*cos(n*x) dx = C - 2*|< 2 2 2 | + |
$$\int - \cos{\left(2 x \right)} \cos{\left(n x \right)}\, dx = C + \begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(n x \right)}}{n} & \text{otherwise} \end{cases} - 2 \left(\begin{cases} - \frac{x \sin^{2}{\left(x \right)} \cos{\left(2 x \right)}}{4} + \frac{x \sin{\left(x \right)} \sin{\left(2 x \right)} \cos{\left(x \right)}}{2} + \frac{x \cos^{2}{\left(x \right)} \cos{\left(2 x \right)}}{4} + \frac{\sin^{2}{\left(x \right)} \sin{\left(2 x \right)}}{2} + \frac{3 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(2 x \right)}}{4} & \text{for}\: n = -2 \\\frac{x \sin^{2}{\left(x \right)}}{2} + \frac{x \cos^{2}{\left(x \right)}}{2} + \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{2} & \text{for}\: n = 0 \\- \frac{x \sin^{2}{\left(x \right)} \cos{\left(2 x \right)}}{4} + \frac{x \sin{\left(x \right)} \sin{\left(2 x \right)} \cos{\left(x \right)}}{2} + \frac{x \cos^{2}{\left(x \right)} \cos{\left(2 x \right)}}{4} + \frac{\sin^{2}{\left(x \right)} \sin{\left(2 x \right)}}{2} + \frac{3 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(2 x \right)}}{4} & \text{for}\: n = 2 \\\frac{n^{2} \sin{\left(n x \right)} \cos^{2}{\left(x \right)}}{n^{3} - 4 n} - \frac{2 n \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(n x \right)}}{n^{3} - 4 n} - \frac{2 \sin^{2}{\left(x \right)} \sin{\left(n x \right)}}{n^{3} - 4 n} - \frac{2 \sin{\left(n x \right)} \cos^{2}{\left(x \right)}}{n^{3} - 4 n} & \text{otherwise} \end{cases}\right)$$
Respuesta
[src]
/ -pi | ---- for Or(n = -2, n = 2) | 4 | | /pi*n\ /3*pi*n\ < 2*cos|----| 2*cos|------| | \ 4 / \ 4 / |- ----------- - ------------- otherwise | 2 2 | -4 + n -4 + n \
$$\begin{cases} - \frac{\pi}{4} & \text{for}\: n = -2 \vee n = 2 \\- \frac{2 \cos{\left(\frac{\pi n}{4} \right)}}{n^{2} - 4} - \frac{2 \cos{\left(\frac{3 \pi n}{4} \right)}}{n^{2} - 4} & \text{otherwise} \end{cases}$$
=
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/ -pi | ---- for Or(n = -2, n = 2) | 4 | | /pi*n\ /3*pi*n\ < 2*cos|----| 2*cos|------| | \ 4 / \ 4 / |- ----------- - ------------- otherwise | 2 2 | -4 + n -4 + n \
$$\begin{cases} - \frac{\pi}{4} & \text{for}\: n = -2 \vee n = 2 \\- \frac{2 \cos{\left(\frac{\pi n}{4} \right)}}{n^{2} - 4} - \frac{2 \cos{\left(\frac{3 \pi n}{4} \right)}}{n^{2} - 4} & \text{otherwise} \end{cases}$$
Piecewise((-pi/4, (n = -2)∨(n = 2)), (-2*cos(pi*n/4)/(-4 + n^2) - 2*cos(3*pi*n/4)/(-4 + n^2), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.