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