Sr Examen
Integral de (x^3-x)*sin(pi*k*x) dx
Solución
Ha introducido
[src]
1 / | | / 3 \ | \x - x/*sin(pi*k*x) dx | / -1
$$\int\limits_{-1}^{1} \left(x^{3} - x\right) \sin{\left(x \pi k \right)}\, dx$$
Integral((x^3 - x)*sin((pi*k)*x), (x, -1, 1))
Respuesta (Indefinida)
[src]
// 0 for k = 0\
|| |
|| // 2 \ |
|| || 2*sin(pi*k*x) x *sin(pi*k*x) 2*x*cos(pi*k*x) | | // 0 for k = 0\
/ || ||- ------------- + -------------- + --------------- for k != 0| | || |
| || || 3 3 pi*k 2 2 | | // 0 for k = 0\ // 0 for k = 0\ || //sin(pi*k*x) \ |
| / 3 \ || || pi *k pi *k | | 3 || | || | || ||----------- for pi*k != 0| |
| \x - x/*sin(pi*k*x) dx = C - 3*|<-|< | | + x *|<-cos(pi*k*x) | - x*|<-cos(pi*k*x) | + |<-|< pi*k | |
| || || 3 | | ||------------- otherwise| ||------------- otherwise| || || | |
/ || || x | | \\ pi*k / \\ pi*k / || \\ x otherwise / |
|| || -- otherwise | | ||------------------------------- otherwise|
|| || 3 | | \\ pi*k /
|| \\ / |
||------------------------------------------------------------------- otherwise|
\\ pi*k /
$$\int \left(x^{3} - x\right) \sin{\left(x \pi k \right)}\, dx = C + x^{3} \left(\begin{cases} 0 & \text{for}\: k = 0 \\- \frac{\cos{\left(\pi k x \right)}}{\pi k} & \text{otherwise} \end{cases}\right) - x \left(\begin{cases} 0 & \text{for}\: k = 0 \\- \frac{\cos{\left(\pi k x \right)}}{\pi k} & \text{otherwise} \end{cases}\right) + \begin{cases} 0 & \text{for}\: k = 0 \\- \frac{\begin{cases} \frac{\sin{\left(\pi k x \right)}}{\pi k} & \text{for}\: \pi k \neq 0 \\x & \text{otherwise} \end{cases}}{\pi k} & \text{otherwise} \end{cases} - 3 \left(\begin{cases} 0 & \text{for}\: k = 0 \\- \frac{\begin{cases} \frac{x^{2} \sin{\left(\pi k x \right)}}{\pi k} + \frac{2 x \cos{\left(\pi k x \right)}}{\pi^{2} k^{2}} - \frac{2 \sin{\left(\pi k x \right)}}{\pi^{3} k^{3}} & \text{for}\: k \neq 0 \\\frac{x^{3}}{3} & \text{otherwise} \end{cases}}{\pi k} & \text{otherwise} \end{cases}\right)$$
Respuesta
[src]
/ 12*sin(pi*k) 4*sin(pi*k) 12*cos(pi*k) |- ------------ + ----------- + ------------ for And(k > -oo, k < oo, k != 0) | 4 4 2 2 3 3 < pi *k pi *k pi *k | | 0 otherwise \
$$\begin{cases} \frac{4 \sin{\left(\pi k \right)}}{\pi^{2} k^{2}} + \frac{12 \cos{\left(\pi k \right)}}{\pi^{3} k^{3}} - \frac{12 \sin{\left(\pi k \right)}}{\pi^{4} k^{4}} & \text{for}\: k > -\infty \wedge k < \infty \wedge k \neq 0 \\0 & \text{otherwise} \end{cases}$$
=
=
/ 12*sin(pi*k) 4*sin(pi*k) 12*cos(pi*k) |- ------------ + ----------- + ------------ for And(k > -oo, k < oo, k != 0) | 4 4 2 2 3 3 < pi *k pi *k pi *k | | 0 otherwise \
$$\begin{cases} \frac{4 \sin{\left(\pi k \right)}}{\pi^{2} k^{2}} + \frac{12 \cos{\left(\pi k \right)}}{\pi^{3} k^{3}} - \frac{12 \sin{\left(\pi k \right)}}{\pi^{4} k^{4}} & \text{for}\: k > -\infty \wedge k < \infty \wedge k \neq 0 \\0 & \text{otherwise} \end{cases}$$
Piecewise((-12*sin(pi*k)/(pi^4*k^4) + 4*sin(pi*k)/(pi^2*k^2) + 12*cos(pi*k)/(pi^3*k^3), (k > -oo)∧(k < oo)∧(Ne(k, 0))), (0, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.