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