Sr Examen
Integral de 2x^4cos(nx) dx
Solución
Ha introducido
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180 / | | 4 | 2*x *cos(n*x) dx | / 0
$$\int\limits_{0}^{180} 2 x^{4} \cos{\left(n x \right)}\, dx$$
Integral((2*x^4)*cos(n*x), (x, 0, 180))
Respuesta (Indefinida)
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// 5 \
|| x |
|| -- for n = 0|
|| 5 |
/ || |
| ||/ 3 2 | // x for n = 0\
| 4 ||| 6*sin(n*x) x *cos(n*x) 3*x *sin(n*x) 6*x*cos(n*x) | 4 || |
| 2*x *cos(n*x) dx = C - 8*|<|- ---------- - ----------- + ------------- + ------------ for n != 0 | + 2*x *|
$$\int 2 x^{4} \cos{\left(n x \right)}\, dx = C + 2 x^{4} \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(n x \right)}}{n} & \text{otherwise} \end{cases}\right) - 8 \left(\begin{cases} \frac{x^{5}}{5} & \text{for}\: n = 0 \\\frac{\begin{cases} - \frac{x^{3} \cos{\left(n x \right)}}{n} + \frac{3 x^{2} \sin{\left(n x \right)}}{n^{2}} + \frac{6 x \cos{\left(n x \right)}}{n^{3}} - \frac{6 \sin{\left(n x \right)}}{n^{4}} & \text{for}\: n \neq 0 \\0 & \text{otherwise} \end{cases}}{n} & \text{otherwise} \end{cases}\right)$$
Respuesta
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/ 777600*sin(180*n) 8640*cos(180*n) 48*sin(180*n) 46656000*cos(180*n) 2099520000*sin(180*n) |- ----------------- - --------------- + ------------- + ------------------- + --------------------- for And(n > -oo, n < oo, n != 0) | 3 4 5 2 n < n n n n | | 75582720000 otherwise \
$$\begin{cases} \frac{2099520000 \sin{\left(180 n \right)}}{n} + \frac{46656000 \cos{\left(180 n \right)}}{n^{2}} - \frac{777600 \sin{\left(180 n \right)}}{n^{3}} - \frac{8640 \cos{\left(180 n \right)}}{n^{4}} + \frac{48 \sin{\left(180 n \right)}}{n^{5}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\75582720000 & \text{otherwise} \end{cases}$$
=
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/ 777600*sin(180*n) 8640*cos(180*n) 48*sin(180*n) 46656000*cos(180*n) 2099520000*sin(180*n) |- ----------------- - --------------- + ------------- + ------------------- + --------------------- for And(n > -oo, n < oo, n != 0) | 3 4 5 2 n < n n n n | | 75582720000 otherwise \
$$\begin{cases} \frac{2099520000 \sin{\left(180 n \right)}}{n} + \frac{46656000 \cos{\left(180 n \right)}}{n^{2}} - \frac{777600 \sin{\left(180 n \right)}}{n^{3}} - \frac{8640 \cos{\left(180 n \right)}}{n^{4}} + \frac{48 \sin{\left(180 n \right)}}{n^{5}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\75582720000 & \text{otherwise} \end{cases}$$
Piecewise((-777600*sin(180*n)/n^3 - 8640*cos(180*n)/n^4 + 48*sin(180*n)/n^5 + 46656000*cos(180*n)/n^2 + 2099520000*sin(180*n)/n, (n > -oo)∧(n < oo)∧(Ne(n, 0))), (75582720000, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.