Sr Examen
Integral de sinax*sinbx dx
Solución
Ha introducido
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1 / | | sin(a*x)*sin(b*x) dx | / 0
$$\int\limits_{0}^{1} \sin{\left(a x \right)} \sin{\left(b x \right)}\, dx$$
Integral(sin(a*x)*sin(b*x), (x, 0, 1))
Respuesta (Indefinida)
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// 0 for And(a = 0, b = 0)\
|| |
|| 2 2 |
|| x*cos (b*x) x*sin (b*x) cos(b*x)*sin(b*x) |
||- ----------- - ----------- + ----------------- for a = -b |
|| 2 2 2*b |
/ || |
| || 2 2 |
| sin(a*x)*sin(b*x) dx = C + |< x*cos (b*x) x*sin (b*x) cos(b*x)*sin(b*x) |
| || ----------- + ----------- - ----------------- for a = b |
/ || 2 2 2*b |
|| |
|| b*cos(b*x)*sin(a*x) a*cos(a*x)*sin(b*x) |
|| ------------------- - ------------------- otherwise |
|| 2 2 2 2 |
|| a - b a - b |
\\ /
$$\int \sin{\left(a x \right)} \sin{\left(b x \right)}\, dx = C + \begin{cases} 0 & \text{for}\: a = 0 \wedge b = 0 \\- \frac{x \sin^{2}{\left(b x \right)}}{2} - \frac{x \cos^{2}{\left(b x \right)}}{2} + \frac{\sin{\left(b x \right)} \cos{\left(b x \right)}}{2 b} & \text{for}\: a = - b \\\frac{x \sin^{2}{\left(b x \right)}}{2} + \frac{x \cos^{2}{\left(b x \right)}}{2} - \frac{\sin{\left(b x \right)} \cos{\left(b x \right)}}{2 b} & \text{for}\: a = b \\- \frac{a \sin{\left(b x \right)} \cos{\left(a x \right)}}{a^{2} - b^{2}} + \frac{b \sin{\left(a x \right)} \cos{\left(b x \right)}}{a^{2} - b^{2}} & \text{otherwise} \end{cases}$$
Respuesta
[src]
/ 0 for Or(And(a = 0, b = 0), And(a = 0, a = b, b = 0), And(a = 0, a = -b, b = 0), And(a = 0, a = -b, a = b, b = 0)) | | 2 2 | cos (b) sin (b) cos(b)*sin(b) |- ------- - ------- + ------------- for Or(And(a = 0, a = -b), And(a = -b, a = b), And(a = -b, b = 0), And(a = 0, a = -b, a = b), And(a = -b, a = b, b = 0), a = -b) | 2 2 2*b | | 2 2 < cos (b) sin (b) cos(b)*sin(b) | ------- + ------- - ------------- for Or(And(a = 0, a = b), And(a = b, b = 0), a = b) | 2 2 2*b | | b*cos(b)*sin(a) a*cos(a)*sin(b) | --------------- - --------------- otherwise | 2 2 2 2 | a - b a - b \
$$\begin{cases} 0 & \text{for}\: \left(a = 0 \wedge b = 0\right) \vee \left(a = 0 \wedge a = b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge a = b \wedge b = 0\right) \\- \frac{\sin^{2}{\left(b \right)}}{2} - \frac{\cos^{2}{\left(b \right)}}{2} + \frac{\sin{\left(b \right)} \cos{\left(b \right)}}{2 b} & \text{for}\: \left(a = 0 \wedge a = - b\right) \vee \left(a = - b \wedge a = b\right) \vee \left(a = - b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge a = b\right) \vee \left(a = - b \wedge a = b \wedge b = 0\right) \vee a = - b \\\frac{\sin^{2}{\left(b \right)}}{2} + \frac{\cos^{2}{\left(b \right)}}{2} - \frac{\sin{\left(b \right)} \cos{\left(b \right)}}{2 b} & \text{for}\: \left(a = 0 \wedge a = b\right) \vee \left(a = b \wedge b = 0\right) \vee a = b \\- \frac{a \sin{\left(b \right)} \cos{\left(a \right)}}{a^{2} - b^{2}} + \frac{b \sin{\left(a \right)} \cos{\left(b \right)}}{a^{2} - b^{2}} & \text{otherwise} \end{cases}$$
=
=
/ 0 for Or(And(a = 0, b = 0), And(a = 0, a = b, b = 0), And(a = 0, a = -b, b = 0), And(a = 0, a = -b, a = b, b = 0)) | | 2 2 | cos (b) sin (b) cos(b)*sin(b) |- ------- - ------- + ------------- for Or(And(a = 0, a = -b), And(a = -b, a = b), And(a = -b, b = 0), And(a = 0, a = -b, a = b), And(a = -b, a = b, b = 0), a = -b) | 2 2 2*b | | 2 2 < cos (b) sin (b) cos(b)*sin(b) | ------- + ------- - ------------- for Or(And(a = 0, a = b), And(a = b, b = 0), a = b) | 2 2 2*b | | b*cos(b)*sin(a) a*cos(a)*sin(b) | --------------- - --------------- otherwise | 2 2 2 2 | a - b a - b \
$$\begin{cases} 0 & \text{for}\: \left(a = 0 \wedge b = 0\right) \vee \left(a = 0 \wedge a = b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge a = b \wedge b = 0\right) \\- \frac{\sin^{2}{\left(b \right)}}{2} - \frac{\cos^{2}{\left(b \right)}}{2} + \frac{\sin{\left(b \right)} \cos{\left(b \right)}}{2 b} & \text{for}\: \left(a = 0 \wedge a = - b\right) \vee \left(a = - b \wedge a = b\right) \vee \left(a = - b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge a = b\right) \vee \left(a = - b \wedge a = b \wedge b = 0\right) \vee a = - b \\\frac{\sin^{2}{\left(b \right)}}{2} + \frac{\cos^{2}{\left(b \right)}}{2} - \frac{\sin{\left(b \right)} \cos{\left(b \right)}}{2 b} & \text{for}\: \left(a = 0 \wedge a = b\right) \vee \left(a = b \wedge b = 0\right) \vee a = b \\- \frac{a \sin{\left(b \right)} \cos{\left(a \right)}}{a^{2} - b^{2}} + \frac{b \sin{\left(a \right)} \cos{\left(b \right)}}{a^{2} - b^{2}} & \text{otherwise} \end{cases}$$
Piecewise((0, ((a = 0)∧(b = 0))∨((a = 0)∧(a = b)∧(b = 0))∨((a = 0)∧(b = 0)∧(a = -b))∨((a = 0)∧(a = b)∧(b = 0)∧(a = -b))), (-cos(b)^2/2 - sin(b)^2/2 + cos(b)*sin(b)/(2*b), (a = -b)∨((a = 0)∧(a = -b))∨((a = b)∧(a = -b))∨((b = 0)∧(a = -b))∨((a = 0)∧(a = b)∧(a = -b))∨((a = b)∧(b = 0)∧(a = -b))), (cos(b)^2/2 + sin(b)^2/2 - cos(b)*sin(b)/(2*b), (a = b)∨((a = 0)∧(a = b))∨((a = b)∧(b = 0))), (b*cos(b)*sin(a)/(a^2 - b^2) - a*cos(a)*sin(b)/(a^2 - b^2), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.