Sr Examen
Integral de (1/pi*500*(3*(r/b)*cos(2*x)+2*sin(3*x)))cos(mx) dx
Solución
Ha introducido
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pi / | | 500 / r \ | ---*|3*-*cos(2*x) + 2*sin(3*x)|*cos(m*x) dx | pi \ b / | / 0
$$\int\limits_{0}^{\pi} \frac{500}{\pi} \left(3 \frac{r}{b} \cos{\left(2 x \right)} + 2 \sin{\left(3 x \right)}\right) \cos{\left(m x \right)}\, dx$$
Integral(((500/pi)*((3*(r/b))*cos(2*x) + 2*sin(3*x)))*cos(m*x), (x, 0, pi))
Respuesta
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/ 0 for Or(m = -3, m = 3) | | /6 3*pi*r\ | 500*|- + ------| | 600 \5 2*b / | --- + ---------------- for Or(m = -2, m = 2) | pi pi < | / 2 \ / 2 3 \ | | 24*b 6*b*m | | 24*b*cos(pi*m) 27*m*r*sin(pi*m) 6*b*m *cos(pi*m) 3*r*m *sin(pi*m) | | 500*|- --------------------- + ---------------------| 500*|--------------------- - --------------------- - --------------------- + ---------------------| | | 4 2 4 2| | 4 2 4 2 4 2 4 2| | \ 36*b + b*m - 13*b*m 36*b + b*m - 13*b*m / \36*b + b*m - 13*b*m 36*b + b*m - 13*b*m 36*b + b*m - 13*b*m 36*b + b*m - 13*b*m / |- ----------------------------------------------------- + --------------------------------------------------------------------------------------------------- otherwise \ pi pi
$$\begin{cases} 0 & \text{for}\: m = -3 \vee m = 3 \\\frac{500 \left(\frac{6}{5} + \frac{3 \pi r}{2 b}\right)}{\pi} + \frac{600}{\pi} & \text{for}\: m = -2 \vee m = 2 \\- \frac{500 \left(\frac{6 b m^{2}}{b m^{4} - 13 b m^{2} + 36 b} - \frac{24 b}{b m^{4} - 13 b m^{2} + 36 b}\right)}{\pi} + \frac{500 \left(- \frac{6 b m^{2} \cos{\left(\pi m \right)}}{b m^{4} - 13 b m^{2} + 36 b} + \frac{24 b \cos{\left(\pi m \right)}}{b m^{4} - 13 b m^{2} + 36 b} + \frac{3 m^{3} r \sin{\left(\pi m \right)}}{b m^{4} - 13 b m^{2} + 36 b} - \frac{27 m r \sin{\left(\pi m \right)}}{b m^{4} - 13 b m^{2} + 36 b}\right)}{\pi} & \text{otherwise} \end{cases}$$
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/ 0 for Or(m = -3, m = 3) | | /6 3*pi*r\ | 500*|- + ------| | 600 \5 2*b / | --- + ---------------- for Or(m = -2, m = 2) | pi pi < | / 2 \ / 2 3 \ | | 24*b 6*b*m | | 24*b*cos(pi*m) 27*m*r*sin(pi*m) 6*b*m *cos(pi*m) 3*r*m *sin(pi*m) | | 500*|- --------------------- + ---------------------| 500*|--------------------- - --------------------- - --------------------- + ---------------------| | | 4 2 4 2| | 4 2 4 2 4 2 4 2| | \ 36*b + b*m - 13*b*m 36*b + b*m - 13*b*m / \36*b + b*m - 13*b*m 36*b + b*m - 13*b*m 36*b + b*m - 13*b*m 36*b + b*m - 13*b*m / |- ----------------------------------------------------- + --------------------------------------------------------------------------------------------------- otherwise \ pi pi
$$\begin{cases} 0 & \text{for}\: m = -3 \vee m = 3 \\\frac{500 \left(\frac{6}{5} + \frac{3 \pi r}{2 b}\right)}{\pi} + \frac{600}{\pi} & \text{for}\: m = -2 \vee m = 2 \\- \frac{500 \left(\frac{6 b m^{2}}{b m^{4} - 13 b m^{2} + 36 b} - \frac{24 b}{b m^{4} - 13 b m^{2} + 36 b}\right)}{\pi} + \frac{500 \left(- \frac{6 b m^{2} \cos{\left(\pi m \right)}}{b m^{4} - 13 b m^{2} + 36 b} + \frac{24 b \cos{\left(\pi m \right)}}{b m^{4} - 13 b m^{2} + 36 b} + \frac{3 m^{3} r \sin{\left(\pi m \right)}}{b m^{4} - 13 b m^{2} + 36 b} - \frac{27 m r \sin{\left(\pi m \right)}}{b m^{4} - 13 b m^{2} + 36 b}\right)}{\pi} & \text{otherwise} \end{cases}$$
Piecewise((0, (m = -3)∨(m = 3)), (600/pi + 500*(6/5 + 3*pi*r/(2*b))/pi, (m = -2)∨(m = 2)), (-500*(-24*b/(36*b + b*m^4 - 13*b*m^2) + 6*b*m^2/(36*b + b*m^4 - 13*b*m^2))/pi + 500*(24*b*cos(pi*m)/(36*b + b*m^4 - 13*b*m^2) - 27*m*r*sin(pi*m)/(36*b + b*m^4 - 13*b*m^2) - 6*b*m^2*cos(pi*m)/(36*b + b*m^4 - 13*b*m^2) + 3*r*m^3*sin(pi*m)/(36*b + b*m^4 - 13*b*m^2))/pi, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.