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