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