Sr Examen
Integral de x*cos(x)*sin(nx) dx
Solución
Ha introducido
[src]
pi / | | x*cos(x)*sin(n*x) dx | / 0
$$\int\limits_{0}^{\pi} x \cos{\left(x \right)} \sin{\left(n x \right)}\, dx$$
Integral((x*cos(x))*sin(n*x), (x, 0, pi))
Respuesta (Indefinida)
[src]
// 2 2 \
|| x*sin (x) cos(x)*sin(x) x*cos (x) |
|| - --------- - ------------- + --------- for n = -1|
|| 4 4 4 |
|| |
/ || 2 2 |
| || x*cos (x) x*sin (x) cos(x)*sin(x) |
| x*cos(x)*sin(n*x) dx = C + |< - --------- + --------- + ------------- for n = 1 |
| || 4 4 4 |
/ || |
|| 2 2 3 |
||cos(x)*sin(n*x) x*sin(x)*sin(n*x) n *cos(x)*sin(n*x) 2*n*cos(n*x)*sin(x) n*x*cos(x)*cos(n*x) x*n *sin(x)*sin(n*x) x*n *cos(x)*cos(n*x) |
||--------------- + ----------------- + ------------------ - ------------------- + ------------------- - -------------------- - -------------------- otherwise |
|| 4 2 4 2 4 2 4 2 4 2 4 2 4 2 |
\\ 1 + n - 2*n 1 + n - 2*n 1 + n - 2*n 1 + n - 2*n 1 + n - 2*n 1 + n - 2*n 1 + n - 2*n /
$$\int x \cos{\left(x \right)} \sin{\left(n x \right)}\, dx = C + \begin{cases} - \frac{x \sin^{2}{\left(x \right)}}{4} + \frac{x \cos^{2}{\left(x \right)}}{4} - \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{4} & \text{for}\: n = -1 \\\frac{x \sin^{2}{\left(x \right)}}{4} - \frac{x \cos^{2}{\left(x \right)}}{4} + \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{4} & \text{for}\: n = 1 \\- \frac{n^{3} x \cos{\left(x \right)} \cos{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} - \frac{n^{2} x \sin{\left(x \right)} \sin{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} + \frac{n^{2} \sin{\left(n x \right)} \cos{\left(x \right)}}{n^{4} - 2 n^{2} + 1} + \frac{n x \cos{\left(x \right)} \cos{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} - \frac{2 n \sin{\left(x \right)} \cos{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} + \frac{x \sin{\left(x \right)} \sin{\left(n x \right)}}{n^{4} - 2 n^{2} + 1} + \frac{\sin{\left(n x \right)} \cos{\left(x \right)}}{n^{4} - 2 n^{2} + 1} & \text{otherwise} \end{cases}$$
Respuesta
[src]
/ pi | -- for n = -1 | 4 | | -pi | ---- for n = 1 < 4 | | 2 3 | sin(pi*n) n *sin(pi*n) pi*n *cos(pi*n) pi*n*cos(pi*n) |- ------------- - ------------- + --------------- - -------------- otherwise | 4 2 4 2 4 2 4 2 \ 1 + n - 2*n 1 + n - 2*n 1 + n - 2*n 1 + n - 2*n
$$\begin{cases} \frac{\pi}{4} & \text{for}\: n = -1 \\- \frac{\pi}{4} & \text{for}\: n = 1 \\\frac{\pi n^{3} \cos{\left(\pi n \right)}}{n^{4} - 2 n^{2} + 1} - \frac{n^{2} \sin{\left(\pi n \right)}}{n^{4} - 2 n^{2} + 1} - \frac{\pi n \cos{\left(\pi n \right)}}{n^{4} - 2 n^{2} + 1} - \frac{\sin{\left(\pi n \right)}}{n^{4} - 2 n^{2} + 1} & \text{otherwise} \end{cases}$$
=
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/ pi | -- for n = -1 | 4 | | -pi | ---- for n = 1 < 4 | | 2 3 | sin(pi*n) n *sin(pi*n) pi*n *cos(pi*n) pi*n*cos(pi*n) |- ------------- - ------------- + --------------- - -------------- otherwise | 4 2 4 2 4 2 4 2 \ 1 + n - 2*n 1 + n - 2*n 1 + n - 2*n 1 + n - 2*n
$$\begin{cases} \frac{\pi}{4} & \text{for}\: n = -1 \\- \frac{\pi}{4} & \text{for}\: n = 1 \\\frac{\pi n^{3} \cos{\left(\pi n \right)}}{n^{4} - 2 n^{2} + 1} - \frac{n^{2} \sin{\left(\pi n \right)}}{n^{4} - 2 n^{2} + 1} - \frac{\pi n \cos{\left(\pi n \right)}}{n^{4} - 2 n^{2} + 1} - \frac{\sin{\left(\pi n \right)}}{n^{4} - 2 n^{2} + 1} & \text{otherwise} \end{cases}$$
Piecewise((pi/4, n = -1), (-pi/4, n = 1), (-sin(pi*n)/(1 + n^4 - 2*n^2) - n^2*sin(pi*n)/(1 + n^4 - 2*n^2) + pi*n^3*cos(pi*n)/(1 + n^4 - 2*n^2) - pi*n*cos(pi*n)/(1 + n^4 - 2*n^2), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.