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