Sr Examen
Integral de te^(-i(2n(pi)t))*cos(2(pi)t) dx
Solución
Ha introducido
[src]
1 / | | -I*2*n*pi*t | t*E *cos(2*pi*t) dt | / 0
$$\int\limits_{0}^{1} e^{- i t \pi 2 n} t \cos{\left(2 \pi t \right)}\, dt$$
Integral((t*E^((-i)*(((2*n)*pi)*t)))*cos((2*pi)*t), (t, 0, 1))
Respuesta (Indefinida)
[src]
/ | 2 2 2 2 2 2 3 3 2 2 3 3 | -I*2*n*pi*t cos(-2*pi*t + 2*pi*n*t) cos(2*pi*t + 2*pi*n*t) n *cos(-2*pi*t + 2*pi*n*t) n *cos(2*pi*t + 2*pi*n*t) I*sin(-2*pi*t + 2*pi*n*t) I*sin(2*pi*t + 2*pi*n*t) 2*n*cos(2*pi*t + 2*pi*n*t) 2*n*cos(-2*pi*t + 2*pi*n*t) I*n *sin(-2*pi*t + 2*pi*n*t) I*n *sin(2*pi*t + 2*pi*n*t) 2*pi*t*sin(-2*pi*t + 2*pi*n*t) 2*I*n*sin(-2*pi*t + 2*pi*n*t) 2*pi*t*sin(2*pi*t + 2*pi*n*t) 2*I*n*sin(2*pi*t + 2*pi*n*t) 2*pi*I*t*cos(-2*pi*t + 2*pi*n*t) 2*pi*n*t*sin(-2*pi*t + 2*pi*n*t) 2*pi*n*t*sin(2*pi*t + 2*pi*n*t) 2*pi*t*n *sin(2*pi*t + 2*pi*n*t) 2*pi*I*t*cos(2*pi*t + 2*pi*n*t) 2*pi*t*n *sin(-2*pi*t + 2*pi*n*t) 2*pi*t*n *sin(-2*pi*t + 2*pi*n*t) 2*pi*t*n *sin(2*pi*t + 2*pi*n*t) 2*pi*I*n*t*cos(-2*pi*t + 2*pi*n*t) 2*pi*I*n*t*cos(2*pi*t + 2*pi*n*t) 2*pi*I*t*n *cos(2*pi*t + 2*pi*n*t) 2*pi*I*t*n *cos(-2*pi*t + 2*pi*n*t) 2*pi*I*t*n *cos(-2*pi*t + 2*pi*n*t) 2*pi*I*t*n *cos(2*pi*t + 2*pi*n*t) | t*E *cos(2*pi*t) dt = C + ---------------------------- + ---------------------------- + ---------------------------- + ---------------------------- - ---------------------------- - ---------------------------- - ---------------------------- + ---------------------------- - ---------------------------- - ---------------------------- - ------------------------------ - ----------------------------- + ----------------------------- + ---------------------------- - -------------------------------- - -------------------------------- - ------------------------------- - -------------------------------- + ------------------------------- + --------------------------------- + --------------------------------- + -------------------------------- - ---------------------------------- - --------------------------------- - ---------------------------------- + ----------------------------------- + ----------------------------------- + ---------------------------------- | 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 / 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n 8*pi - 16*pi *n + 8*pi *n
$$\int e^{- i t \pi 2 n} t \cos{\left(2 \pi t \right)}\, dt = C + \frac{2 \pi n^{3} t \sin{\left(2 \pi n t - 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} + \frac{2 \pi n^{3} t \sin{\left(2 \pi n t + 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} + \frac{2 i \pi n^{3} t \cos{\left(2 \pi n t - 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} + \frac{2 i \pi n^{3} t \cos{\left(2 \pi n t + 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} + \frac{2 \pi n^{2} t \sin{\left(2 \pi n t - 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} - \frac{2 \pi n^{2} t \sin{\left(2 \pi n t + 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} + \frac{2 i \pi n^{2} t \cos{\left(2 \pi n t - 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} - \frac{2 i \pi n^{2} t \cos{\left(2 \pi n t + 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} - \frac{i n^{2} \sin{\left(2 \pi n t - 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} - \frac{i n^{2} \sin{\left(2 \pi n t + 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} + \frac{n^{2} \cos{\left(2 \pi n t - 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} + \frac{n^{2} \cos{\left(2 \pi n t + 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} - \frac{2 \pi n t \sin{\left(2 \pi n t - 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} - \frac{2 \pi n t \sin{\left(2 \pi n t + 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} - \frac{2 i \pi n t \cos{\left(2 \pi n t - 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} - \frac{2 i \pi n t \cos{\left(2 \pi n t + 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} - \frac{2 i n \sin{\left(2 \pi n t - 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} + \frac{2 i n \sin{\left(2 \pi n t + 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} + \frac{2 n \cos{\left(2 \pi n t - 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} - \frac{2 n \cos{\left(2 \pi n t + 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} - \frac{2 \pi t \sin{\left(2 \pi n t - 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} + \frac{2 \pi t \sin{\left(2 \pi n t + 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} - \frac{2 i \pi t \cos{\left(2 \pi n t - 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} + \frac{2 i \pi t \cos{\left(2 \pi n t + 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} - \frac{i \sin{\left(2 \pi n t - 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} - \frac{i \sin{\left(2 \pi n t + 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} + \frac{\cos{\left(2 \pi n t - 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}} + \frac{\cos{\left(2 \pi n t + 2 \pi t \right)}}{8 \pi^{2} n^{4} - 16 \pi^{2} n^{2} + 8 \pi^{2}}$$
Respuesta
[src]
/ 1 I | - - ---- for n = -1 | 4 8*pi | | 1 I | - + ---- for n = 1 < 4 8*pi | | 2 2 3 | 1 1 n n 2*pi*I*n 2*pi*I*n |--------------------------------------------------------- - --------------------------- + --------------------------------------------------------- - --------------------------- - --------------------------------------------------------- + --------------------------------------------------------- otherwise | 2 2*pi*I*n 2 2 2*pi*I*n 2 4 2*pi*I*n 2 2 2 2 4 2 2*pi*I*n 2 2 2*pi*I*n 2 4 2*pi*I*n 2 2 2 2 4 2 2*pi*I*n 2 2 2*pi*I*n 2 4 2*pi*I*n 2 2*pi*I*n 2 2 2*pi*I*n 2 4 2*pi*I*n \4*pi *e - 8*pi *n *e + 4*pi *n *e 4*pi - 8*pi *n + 4*pi *n 4*pi *e - 8*pi *n *e + 4*pi *n *e 4*pi - 8*pi *n + 4*pi *n 4*pi *e - 8*pi *n *e + 4*pi *n *e 4*pi *e - 8*pi *n *e + 4*pi *n *e
$$\begin{cases} \frac{1}{4} - \frac{i}{8 \pi} & \text{for}\: n = -1 \\\frac{1}{4} + \frac{i}{8 \pi} & \text{for}\: n = 1 \\\frac{2 i \pi n^{3}}{4 \pi^{2} n^{4} e^{2 i \pi n} - 8 \pi^{2} n^{2} e^{2 i \pi n} + 4 \pi^{2} e^{2 i \pi n}} + \frac{n^{2}}{4 \pi^{2} n^{4} e^{2 i \pi n} - 8 \pi^{2} n^{2} e^{2 i \pi n} + 4 \pi^{2} e^{2 i \pi n}} - \frac{n^{2}}{4 \pi^{2} n^{4} - 8 \pi^{2} n^{2} + 4 \pi^{2}} - \frac{2 i \pi n}{4 \pi^{2} n^{4} e^{2 i \pi n} - 8 \pi^{2} n^{2} e^{2 i \pi n} + 4 \pi^{2} e^{2 i \pi n}} + \frac{1}{4 \pi^{2} n^{4} e^{2 i \pi n} - 8 \pi^{2} n^{2} e^{2 i \pi n} + 4 \pi^{2} e^{2 i \pi n}} - \frac{1}{4 \pi^{2} n^{4} - 8 \pi^{2} n^{2} + 4 \pi^{2}} & \text{otherwise} \end{cases}$$
=
=
/ 1 I | - - ---- for n = -1 | 4 8*pi | | 1 I | - + ---- for n = 1 < 4 8*pi | | 2 2 3 | 1 1 n n 2*pi*I*n 2*pi*I*n |--------------------------------------------------------- - --------------------------- + --------------------------------------------------------- - --------------------------- - --------------------------------------------------------- + --------------------------------------------------------- otherwise | 2 2*pi*I*n 2 2 2*pi*I*n 2 4 2*pi*I*n 2 2 2 2 4 2 2*pi*I*n 2 2 2*pi*I*n 2 4 2*pi*I*n 2 2 2 2 4 2 2*pi*I*n 2 2 2*pi*I*n 2 4 2*pi*I*n 2 2*pi*I*n 2 2 2*pi*I*n 2 4 2*pi*I*n \4*pi *e - 8*pi *n *e + 4*pi *n *e 4*pi - 8*pi *n + 4*pi *n 4*pi *e - 8*pi *n *e + 4*pi *n *e 4*pi - 8*pi *n + 4*pi *n 4*pi *e - 8*pi *n *e + 4*pi *n *e 4*pi *e - 8*pi *n *e + 4*pi *n *e
$$\begin{cases} \frac{1}{4} - \frac{i}{8 \pi} & \text{for}\: n = -1 \\\frac{1}{4} + \frac{i}{8 \pi} & \text{for}\: n = 1 \\\frac{2 i \pi n^{3}}{4 \pi^{2} n^{4} e^{2 i \pi n} - 8 \pi^{2} n^{2} e^{2 i \pi n} + 4 \pi^{2} e^{2 i \pi n}} + \frac{n^{2}}{4 \pi^{2} n^{4} e^{2 i \pi n} - 8 \pi^{2} n^{2} e^{2 i \pi n} + 4 \pi^{2} e^{2 i \pi n}} - \frac{n^{2}}{4 \pi^{2} n^{4} - 8 \pi^{2} n^{2} + 4 \pi^{2}} - \frac{2 i \pi n}{4 \pi^{2} n^{4} e^{2 i \pi n} - 8 \pi^{2} n^{2} e^{2 i \pi n} + 4 \pi^{2} e^{2 i \pi n}} + \frac{1}{4 \pi^{2} n^{4} e^{2 i \pi n} - 8 \pi^{2} n^{2} e^{2 i \pi n} + 4 \pi^{2} e^{2 i \pi n}} - \frac{1}{4 \pi^{2} n^{4} - 8 \pi^{2} n^{2} + 4 \pi^{2}} & \text{otherwise} \end{cases}$$
Piecewise((1/4 - i/(8*pi), n = -1), (1/4 + i/(8*pi), n = 1), (1/(4*pi^2*exp(2*pi*i*n) - 8*pi^2*n^2*exp(2*pi*i*n) + 4*pi^2*n^4*exp(2*pi*i*n)) - 1/(4*pi^2 - 8*pi^2*n^2 + 4*pi^2*n^4) + n^2/(4*pi^2*exp(2*pi*i*n) - 8*pi^2*n^2*exp(2*pi*i*n) + 4*pi^2*n^4*exp(2*pi*i*n)) - n^2/(4*pi^2 - 8*pi^2*n^2 + 4*pi^2*n^4) - 2*pi*i*n/(4*pi^2*exp(2*pi*i*n) - 8*pi^2*n^2*exp(2*pi*i*n) + 4*pi^2*n^4*exp(2*pi*i*n)) + 2*pi*i*n^3/(4*pi^2*exp(2*pi*i*n) - 8*pi^2*n^2*exp(2*pi*i*n) + 4*pi^2*n^4*exp(2*pi*i*n)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.