Sr Examen
Integral de cos(πx/2l)x dx
Solución
Ha introducido
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0 / | | /pi*x \ | cos|----*l|*x dx | \ 2 / | / l - 2
$$\int\limits_{\frac{l}{2}}^{0} x \cos{\left(l \frac{\pi x}{2} \right)}\, dx$$
Integral(cos(((pi*x)/2)*l)*x, (x, l/2, 0))
Respuesta (Indefinida)
[src]
// 2 \
|| x |
|| -- for l = 0|
|| 2 |
/ || | // x for l = 0\
| || // /pi*l*x\ \ | || |
| /pi*x \ || ||-2*cos|------| | | || /pi*l*x\ |
| cos|----*l|*x dx = C - |< || \ 2 / pi*l | | + x*|<2*sin|------| |
| \ 2 / ||2*|<-------------- for ---- != 0| | || \ 2 / |
| || || pi*l 2 | | ||------------- otherwise|
/ || || | | \\ pi*l /
|| \\ 0 otherwise / |
||---------------------------------- otherwise|
|| pi*l |
\\ /
$$\int x \cos{\left(l \frac{\pi x}{2} \right)}\, dx = C + x \left(\begin{cases} x & \text{for}\: l = 0 \\\frac{2 \sin{\left(\frac{\pi l x}{2} \right)}}{\pi l} & \text{otherwise} \end{cases}\right) - \begin{cases} \frac{x^{2}}{2} & \text{for}\: l = 0 \\\frac{2 \left(\begin{cases} - \frac{2 \cos{\left(\frac{\pi l x}{2} \right)}}{\pi l} & \text{for}\: \frac{\pi l}{2} \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{\pi l} & \text{otherwise} \end{cases}$$
Respuesta
[src]
/ / 2\ / 2\ | |pi*l | |pi*l | | sin|-----| 4*cos|-----| | \ 4 / 4 \ 4 / |- ---------- + ------ - ------------ for And(l > -oo, l < oo, l != 0) | pi 2 2 2 2 < pi *l pi *l | | 2 | -l | ---- otherwise | 8 \
$$\begin{cases} - \frac{\sin{\left(\frac{\pi l^{2}}{4} \right)}}{\pi} - \frac{4 \cos{\left(\frac{\pi l^{2}}{4} \right)}}{\pi^{2} l^{2}} + \frac{4}{\pi^{2} l^{2}} & \text{for}\: l > -\infty \wedge l < \infty \wedge l \neq 0 \\- \frac{l^{2}}{8} & \text{otherwise} \end{cases}$$
=
=
/ / 2\ / 2\ | |pi*l | |pi*l | | sin|-----| 4*cos|-----| | \ 4 / 4 \ 4 / |- ---------- + ------ - ------------ for And(l > -oo, l < oo, l != 0) | pi 2 2 2 2 < pi *l pi *l | | 2 | -l | ---- otherwise | 8 \
$$\begin{cases} - \frac{\sin{\left(\frac{\pi l^{2}}{4} \right)}}{\pi} - \frac{4 \cos{\left(\frac{\pi l^{2}}{4} \right)}}{\pi^{2} l^{2}} + \frac{4}{\pi^{2} l^{2}} & \text{for}\: l > -\infty \wedge l < \infty \wedge l \neq 0 \\- \frac{l^{2}}{8} & \text{otherwise} \end{cases}$$
Piecewise((-sin(pi*l^2/4)/pi + 4/(pi^2*l^2) - 4*cos(pi*l^2/4)/(pi^2*l^2), (l > -oo)∧(l < oo)∧(Ne(l, 0))), (-l^2/8, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.