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