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