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