Sr Examen
Integral de (0,3*x^3+2)*cos(n*x) dx
Solución
Ha introducido
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1 / | | / 3 \ | |3*x | | |---- + 2|*cos(n*x) dx | \ 10 / | / 0
$$\int\limits_{0}^{1} \left(\frac{3 x^{3}}{10} + 2\right) \cos{\left(n x \right)}\, dx$$
Integral((3*x^3/10 + 2)*cos(n*x), (x, 0, 1))
Respuesta (Indefinida)
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// 4 \
|| x |
|| -- for n = 0|
|| 4 |
|| |
||/ 2 |
|||2*cos(n*x) x *cos(n*x) 2*x*sin(n*x) |
9*|<|---------- - ----------- + ------------ for n != 0 |
||< 3 n 2 |
||| n n |
||| | // x for n = 0\
/ ||\ 0 otherwise | 3 || |
| ||---------------------------------------------------- otherwise| 3*x *|
$$\int \left(\frac{3 x^{3}}{10} + 2\right) \cos{\left(n x \right)}\, dx = C + \frac{3 x^{3} \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(n x \right)}}{n} & \text{otherwise} \end{cases}\right)}{10} + 2 \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(n x \right)}}{n} & \text{otherwise} \end{cases}\right) - \frac{9 \left(\begin{cases} \frac{x^{4}}{4} & \text{for}\: n = 0 \\\frac{\begin{cases} - \frac{x^{2} \cos{\left(n x \right)}}{n} + \frac{2 x \sin{\left(n x \right)}}{n^{2}} + \frac{2 \cos{\left(n x \right)}}{n^{3}} & \text{for}\: n \neq 0 \\0 & \text{otherwise} \end{cases}}{n} & \text{otherwise} \end{cases}\right)}{10}$$
Respuesta
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/ 9 9*cos(n) 9*sin(n) 9*cos(n) 23*sin(n) |---- - -------- - -------- + -------- + --------- for And(n > -oo, n < oo, n != 0) | 4 4 3 2 10*n |5*n 5*n 5*n 10*n < | 83 | -- otherwise | 40 \
$$\begin{cases} \frac{23 \sin{\left(n \right)}}{10 n} + \frac{9 \cos{\left(n \right)}}{10 n^{2}} - \frac{9 \sin{\left(n \right)}}{5 n^{3}} - \frac{9 \cos{\left(n \right)}}{5 n^{4}} + \frac{9}{5 n^{4}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\frac{83}{40} & \text{otherwise} \end{cases}$$
=
=
/ 9 9*cos(n) 9*sin(n) 9*cos(n) 23*sin(n) |---- - -------- - -------- + -------- + --------- for And(n > -oo, n < oo, n != 0) | 4 4 3 2 10*n |5*n 5*n 5*n 10*n < | 83 | -- otherwise | 40 \
$$\begin{cases} \frac{23 \sin{\left(n \right)}}{10 n} + \frac{9 \cos{\left(n \right)}}{10 n^{2}} - \frac{9 \sin{\left(n \right)}}{5 n^{3}} - \frac{9 \cos{\left(n \right)}}{5 n^{4}} + \frac{9}{5 n^{4}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\frac{83}{40} & \text{otherwise} \end{cases}$$
Piecewise((9/(5*n^4) - 9*cos(n)/(5*n^4) - 9*sin(n)/(5*n^3) + 9*cos(n)/(10*n^2) + 23*sin(n)/(10*n), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (83/40, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.