Sr Examen
Integral de xcos(pi(2k+1)/(2l)x) dx
Solución
Ha introducido
[src]
1 / | | /pi*(2*k + 1) \ | x*cos|------------*x| dx | \ 2*l / | / 0
$$\int\limits_{0}^{1} x \cos{\left(x \frac{\pi \left(2 k + 1\right)}{2 l} \right)}\, dx$$
Integral(x*cos(((pi*(2*k + 1))/((2*l)))*x), (x, 0, 1))
Respuesta (Indefinida)
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/ 2 /pi*x pi*k*x\ /pi*x pi*k*x\ /pi*x pi*k*x\ | 4*l *cos|---- + ------| 2*pi*l*x*sin|---- + ------| 4*pi*k*l*x*sin|---- + ------| | /pi*(2*k + 1) \ \2*l l / \2*l l / \2*l l / | x*cos|------------*x| dx = C + ------------------------ + --------------------------- + ----------------------------- | \ 2*l / 2 2 2 2 2 2 2 2 2 2 2 2 | pi + 4*k*pi + 4*pi *k pi + 4*k*pi + 4*pi *k pi + 4*k*pi + 4*pi *k /
$$\int x \cos{\left(x \frac{\pi \left(2 k + 1\right)}{2 l} \right)}\, dx = C + \frac{4 \pi k l x \sin{\left(\frac{\pi k x}{l} + \frac{\pi x}{2 l} \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} + \frac{4 l^{2} \cos{\left(\frac{\pi k x}{l} + \frac{\pi x}{2 l} \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} + \frac{2 \pi l x \sin{\left(\frac{\pi k x}{l} + \frac{\pi x}{2 l} \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}}$$
Respuesta
[src]
2 / pi pi*k\ / pi pi*k\ / pi pi*k\
2 4*l *cos|--- + ----| 2*pi*l*sin|--- + ----| 4*pi*k*l*sin|--- + ----|
4*l \2*l l / \2*l l / \2*l l /
- ------------------------ + ------------------------ + ------------------------ + ------------------------
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
pi + 4*k*pi + 4*pi *k pi + 4*k*pi + 4*pi *k pi + 4*k*pi + 4*pi *k pi + 4*k*pi + 4*pi *k
$$\frac{4 \pi k l \sin{\left(\frac{\pi k}{l} + \frac{\pi}{2 l} \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} + \frac{4 l^{2} \cos{\left(\frac{\pi k}{l} + \frac{\pi}{2 l} \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} - \frac{4 l^{2}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} + \frac{2 \pi l \sin{\left(\frac{\pi k}{l} + \frac{\pi}{2 l} \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}}$$
=
=
2 / pi pi*k\ / pi pi*k\ / pi pi*k\
2 4*l *cos|--- + ----| 2*pi*l*sin|--- + ----| 4*pi*k*l*sin|--- + ----|
4*l \2*l l / \2*l l / \2*l l /
- ------------------------ + ------------------------ + ------------------------ + ------------------------
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
pi + 4*k*pi + 4*pi *k pi + 4*k*pi + 4*pi *k pi + 4*k*pi + 4*pi *k pi + 4*k*pi + 4*pi *k
$$\frac{4 \pi k l \sin{\left(\frac{\pi k}{l} + \frac{\pi}{2 l} \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} + \frac{4 l^{2} \cos{\left(\frac{\pi k}{l} + \frac{\pi}{2 l} \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} - \frac{4 l^{2}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} + \frac{2 \pi l \sin{\left(\frac{\pi k}{l} + \frac{\pi}{2 l} \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}}$$
-4*l^2/(pi^2 + 4*k*pi^2 + 4*pi^2*k^2) + 4*l^2*cos(pi/(2*l) + pi*k/l)/(pi^2 + 4*k*pi^2 + 4*pi^2*k^2) + 2*pi*l*sin(pi/(2*l) + pi*k/l)/(pi^2 + 4*k*pi^2 + 4*pi^2*k^2) + 4*pi*k*l*sin(pi/(2*l) + pi*k/l)/(pi^2 + 4*k*pi^2 + 4*pi^2*k^2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.