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