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Resolución de integrales/ xcos(pi(2k+1)/(2l)x)

Integral de xcos(pi(2k+1)/(2l)x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

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) [src] 
  /                                  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   
/
$$\int x \cos{\left(x \frac{\pi \left(2 k + 1\right)}{2 l} \right)}\, dx = C + \frac{4 \pi k l x \sin{\left(\frac{\pi k x}{l} + \frac{\pi x}{2 l} \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} + \frac{4 l^{2} \cos{\left(\frac{\pi k x}{l} + \frac{\pi x}{2 l} \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}} + \frac{2 \pi l x \sin{\left(\frac{\pi k x}{l} + \frac{\pi x}{2 l} \right)}}{4 \pi^{2} k^{2} + 4 \pi^{2} k + \pi^{2}}$$
Simplificar
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.

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