Sr Examen
Integral de (x^2-0.5*x)*sin(2*pi*n*x) dx
Solución
Ha introducido
[src]
1/2 / | | / 2 x\ | |x - -|*sin(2*pi*n*x) dx | \ 2/ | / 0
$$\int\limits_{0}^{\frac{1}{2}} \left(x^{2} - \frac{x}{2}\right) \sin{\left(x 2 \pi n \right)}\, dx$$
Integral((x^2 - x/2)*sin(((2*pi)*n)*x), (x, 0, 1/2))
Respuesta (Indefinida)
[src]
/ 0 for n = 0
|
| //sin(2*pi*n*x) \
| ||------------- for 2*pi*n != 0| // 0 for n = 0\
<-|< 2*pi*n | || | // 0 for n = 0\
| || | || //cos(2*pi*n*x) x*sin(2*pi*n*x) \ | || |
/ | \\ x otherwise / || ||------------- + --------------- for n != 0| | x*|<-cos(2*pi*n*x) |
| |----------------------------------- otherwise || || 2 2 2*pi*n | | // 0 for n = 0\ ||--------------- otherwise|
| / 2 x\ \ 2*pi*n || || 4*pi *n | | 2 || | \\ 2*pi*n /
| |x - -|*sin(2*pi*n*x) dx = C + ----------------------------------------------- - 2*|<-|< | | + x *|<-cos(2*pi*n*x) | - -------------------------------
| \ 2/ 2 || || 2 | | ||--------------- otherwise| 2
| || || x | | \\ 2*pi*n /
/ || || -- otherwise | |
|| \\ 2 / |
||------------------------------------------------ otherwise|
\\ 2*pi*n /
$$\int \left(x^{2} - \frac{x}{2}\right) \sin{\left(x 2 \pi n \right)}\, dx = C + x^{2} \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\cos{\left(2 \pi n x \right)}}{2 \pi n} & \text{otherwise} \end{cases}\right) - \frac{x \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\cos{\left(2 \pi n x \right)}}{2 \pi n} & \text{otherwise} \end{cases}\right)}{2} + \frac{\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\begin{cases} \frac{\sin{\left(2 \pi n x \right)}}{2 \pi n} & \text{for}\: 2 \pi n \neq 0 \\x & \text{otherwise} \end{cases}}{2 \pi n} & \text{otherwise} \end{cases}}{2} - 2 \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\begin{cases} \frac{x \sin{\left(2 \pi n x \right)}}{2 \pi n} + \frac{\cos{\left(2 \pi n x \right)}}{4 \pi^{2} n^{2}} & \text{for}\: n \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases}}{2 \pi n} & \text{otherwise} \end{cases}\right)$$
Respuesta
[src]
/ 1 cos(pi*n) sin(pi*n) |- -------- + --------- + --------- for And(n > -oo, n < oo, n != 0) | 3 3 3 3 2 2 < 4*pi *n 4*pi *n 8*pi *n | | 0 otherwise \
$$\begin{cases} \frac{\sin{\left(\pi n \right)}}{8 \pi^{2} n^{2}} + \frac{\cos{\left(\pi n \right)}}{4 \pi^{3} n^{3}} - \frac{1}{4 \pi^{3} n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases}$$
=
=
/ 1 cos(pi*n) sin(pi*n) |- -------- + --------- + --------- for And(n > -oo, n < oo, n != 0) | 3 3 3 3 2 2 < 4*pi *n 4*pi *n 8*pi *n | | 0 otherwise \
$$\begin{cases} \frac{\sin{\left(\pi n \right)}}{8 \pi^{2} n^{2}} + \frac{\cos{\left(\pi n \right)}}{4 \pi^{3} n^{3}} - \frac{1}{4 \pi^{3} n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases}$$
Piecewise((-1/(4*pi^3*n^3) + cos(pi*n)/(4*pi^3*n^3) + sin(pi*n)/(8*pi^2*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.