Sr Examen
Integral de (e^(a*x)-sin(b*x))^2 dx
Solución
Ha introducido
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1 / | | 2 | / a*x \ | \E - sin(b*x)/ dx | / 0
$$\int\limits_{0}^{1} \left(e^{a x} - \sin{\left(b x \right)}\right)^{2}\, dx$$
Integral((E^(a*x) - sin(b*x))^2, (x, 0, 1))
Respuesta (Indefinida)
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// /cos(b*x) x*sin(b*x) \
|| |-------- + ---------- for b != 0 |
|| | 2 b |
|| | b |
|| < for a = 0|
|| | 2 |
|| | x |
|| | -- otherwise |
|| \ 2 |
/ x for b = 0 || |
/ | ||/ x for And(a = 0, b = 0) |
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$$\int \left(e^{a x} - \sin{\left(b x \right)}\right)^{2}\, dx = C + 2 b \left(\begin{cases} \begin{cases} \frac{x \sin{\left(b x \right)}}{b} + \frac{\cos{\left(b x \right)}}{b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases} & \text{for}\: a = 0 \\\frac{\begin{cases} x & \text{for}\: a = 0 \wedge b = 0 \\\frac{i x e^{- i b x} \sin{\left(b x \right)}}{2} + \frac{x e^{- i b x} \cos{\left(b x \right)}}{2} + \frac{e^{- i b x} \sin{\left(b x \right)}}{2 b} & \text{for}\: a = - i b \\- \frac{i x e^{i b x} \sin{\left(b x \right)}}{2} + \frac{x e^{i b x} \cos{\left(b x \right)}}{2} + \frac{e^{i b x} \sin{\left(b x \right)}}{2 b} & \text{for}\: a = i b \\\frac{a e^{a x} \cos{\left(b x \right)}}{a^{2} + b^{2}} + \frac{b e^{a x} \sin{\left(b x \right)}}{a^{2} + b^{2}} & \text{otherwise} \end{cases}}{a} & \text{otherwise} \end{cases}\right) + \frac{x}{2} - 2 \left(\begin{cases} x & \text{for}\: a = 0 \\\frac{e^{a x}}{a} & \text{otherwise} \end{cases}\right) \sin{\left(b x \right)} + \begin{cases} x & \text{for}\: a = 0 \\\frac{e^{2 a x}}{2 a} & \text{otherwise} \end{cases} - \frac{\begin{cases} x & \text{for}\: b = 0 \\\frac{\sin{\left(2 b x \right)}}{2 b} & \text{otherwise} \end{cases}}{2}$$
Respuesta
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/ 1 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 2 | cos (b) sin (b) 2 2*cos(b) cos(b)*sin(b) | 1 + ------- + ------- - - + -------- - ------------- for Or(And(a = 0, a = -I*b), And(a = 0, a = I*b), And(a = 0, a = -I*b, a = I*b), a = 0) | 2 2 b b 2*b | | 2 2 -2*I*b -I*b | cos (b) sin (b) -I*b I -I*b I*e cos(b)*sin(b) I*e *sin(b) | ------- + ------- - e *sin(b) - --- + I*cos(b)*e + --------- - ------------- - -------------- for Or(And(a = -I*b, a = I*b), And(a = -I*b, b = 0), And(a = -I*b, a = I*b, b = 0), a = -I*b) | 2 2 2*b 2*b 2*b b | | 2 2 2*I*b I*b < cos (b) sin (b) I I*b I*b I*e cos(b)*sin(b) I*e *sin(b) | ------- + ------- + --- - e *sin(b) - I*cos(b)*e - -------- - ------------- + ------------- for Or(And(a = I*b, b = 0), a = I*b) | 2 2 2*b 2*b 2*b b | | 2*a | 1 e | - --- + ---- for b = 0 | 2*a 2*a | | 3 3 2*a 2 2 3 2 3 2 2 2*a 3 2 3 2 3 2 2 a 2 a | b b *e b*a 4*a*b a*b *cos (b) a*b *sin (b) b*a *e b*a *cos (b) b*a *sin (b) a *cos(b)*sin(b) a*b *cos(b)*sin(b) 4*b*a *e *sin(b) 4*a*b *cos(b)*e |- --------------- + --------------- - --------------- - --------------- + --------------- + --------------- + --------------- + --------------- + --------------- - ---------------- - ------------------ - ---------------- + ---------------- otherwise | 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 \ 2*a*b + 2*b*a 2*a*b + 2*b*a 2*a*b + 2*b*a 2*a*b + 2*b*a 2*a*b + 2*b*a 2*a*b + 2*b*a 2*a*b + 2*b*a 2*a*b + 2*b*a 2*a*b + 2*b*a 2*a*b + 2*b*a 2*a*b + 2*b*a 2*a*b + 2*b*a 2*a*b + 2*b*a
$$\begin{cases} 1 & \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{\sin^{2}{\left(b \right)}}{2} + \frac{\cos^{2}{\left(b \right)}}{2} + 1 - \frac{\sin{\left(b \right)} \cos{\left(b \right)}}{2 b} + \frac{2 \cos{\left(b \right)}}{b} - \frac{2}{b} & \text{for}\: \left(a = 0 \wedge a = - i b\right) \vee \left(a = 0 \wedge a = i b\right) \vee \left(a = 0 \wedge a = - i b \wedge a = i b\right) \vee a = 0 \\\frac{\sin^{2}{\left(b \right)}}{2} + \frac{\cos^{2}{\left(b \right)}}{2} - e^{- i b} \sin{\left(b \right)} + i e^{- i b} \cos{\left(b \right)} - \frac{\sin{\left(b \right)} \cos{\left(b \right)}}{2 b} - \frac{i}{2 b} - \frac{i e^{- i b} \sin{\left(b \right)}}{b} + \frac{i e^{- 2 i b}}{2 b} & \text{for}\: \left(a = - i b \wedge a = i b\right) \vee \left(a = - i b \wedge b = 0\right) \vee \left(a = - i b \wedge a = i b \wedge b = 0\right) \vee a = - i b \\- e^{i b} \sin{\left(b \right)} - i e^{i b} \cos{\left(b \right)} + \frac{\sin^{2}{\left(b \right)}}{2} + \frac{\cos^{2}{\left(b \right)}}{2} - \frac{i e^{2 i b}}{2 b} + \frac{i e^{i b} \sin{\left(b \right)}}{b} - \frac{\sin{\left(b \right)} \cos{\left(b \right)}}{2 b} + \frac{i}{2 b} & \text{for}\: \left(a = i b \wedge b = 0\right) \vee a = i b \\\frac{e^{2 a}}{2 a} - \frac{1}{2 a} & \text{for}\: b = 0 \\\frac{a^{3} b \sin^{2}{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} + \frac{a^{3} b \cos^{2}{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} - \frac{a^{3} \sin{\left(b \right)} \cos{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} + \frac{a^{2} b e^{2 a}}{2 a^{3} b + 2 a b^{3}} - \frac{4 a^{2} b e^{a} \sin{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} - \frac{a^{2} b}{2 a^{3} b + 2 a b^{3}} + \frac{a b^{3} \sin^{2}{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} + \frac{a b^{3} \cos^{2}{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} + \frac{4 a b^{2} e^{a} \cos{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} - \frac{a b^{2} \sin{\left(b \right)} \cos{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} - \frac{4 a b^{2}}{2 a^{3} b + 2 a b^{3}} + \frac{b^{3} e^{2 a}}{2 a^{3} b + 2 a b^{3}} - \frac{b^{3}}{2 a^{3} b + 2 a b^{3}} & \text{otherwise} \end{cases}$$
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/ 1 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 2 | cos (b) sin (b) 2 2*cos(b) cos(b)*sin(b) | 1 + ------- + ------- - - + -------- - ------------- for Or(And(a = 0, a = -I*b), And(a = 0, a = I*b), And(a = 0, a = -I*b, a = I*b), a = 0) | 2 2 b b 2*b | | 2 2 -2*I*b -I*b | cos (b) sin (b) -I*b I -I*b I*e cos(b)*sin(b) I*e *sin(b) | ------- + ------- - e *sin(b) - --- + I*cos(b)*e + --------- - ------------- - -------------- for Or(And(a = -I*b, a = I*b), And(a = -I*b, b = 0), And(a = -I*b, a = I*b, b = 0), a = -I*b) | 2 2 2*b 2*b 2*b b | | 2 2 2*I*b I*b < cos (b) sin (b) I I*b I*b I*e cos(b)*sin(b) I*e *sin(b) | ------- + ------- + --- - e *sin(b) - I*cos(b)*e - -------- - ------------- + ------------- for Or(And(a = I*b, b = 0), a = I*b) | 2 2 2*b 2*b 2*b b | | 2*a | 1 e | - --- + ---- for b = 0 | 2*a 2*a | | 3 3 2*a 2 2 3 2 3 2 2 2*a 3 2 3 2 3 2 2 a 2 a | b b *e b*a 4*a*b a*b *cos (b) a*b *sin (b) b*a *e b*a *cos (b) b*a *sin (b) a *cos(b)*sin(b) a*b *cos(b)*sin(b) 4*b*a *e *sin(b) 4*a*b *cos(b)*e |- --------------- + --------------- - --------------- - --------------- + --------------- + --------------- + --------------- + --------------- + --------------- - ---------------- - ------------------ - ---------------- + ---------------- otherwise | 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 \ 2*a*b + 2*b*a 2*a*b + 2*b*a 2*a*b + 2*b*a 2*a*b + 2*b*a 2*a*b + 2*b*a 2*a*b + 2*b*a 2*a*b + 2*b*a 2*a*b + 2*b*a 2*a*b + 2*b*a 2*a*b + 2*b*a 2*a*b + 2*b*a 2*a*b + 2*b*a 2*a*b + 2*b*a
$$\begin{cases} 1 & \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{\sin^{2}{\left(b \right)}}{2} + \frac{\cos^{2}{\left(b \right)}}{2} + 1 - \frac{\sin{\left(b \right)} \cos{\left(b \right)}}{2 b} + \frac{2 \cos{\left(b \right)}}{b} - \frac{2}{b} & \text{for}\: \left(a = 0 \wedge a = - i b\right) \vee \left(a = 0 \wedge a = i b\right) \vee \left(a = 0 \wedge a = - i b \wedge a = i b\right) \vee a = 0 \\\frac{\sin^{2}{\left(b \right)}}{2} + \frac{\cos^{2}{\left(b \right)}}{2} - e^{- i b} \sin{\left(b \right)} + i e^{- i b} \cos{\left(b \right)} - \frac{\sin{\left(b \right)} \cos{\left(b \right)}}{2 b} - \frac{i}{2 b} - \frac{i e^{- i b} \sin{\left(b \right)}}{b} + \frac{i e^{- 2 i b}}{2 b} & \text{for}\: \left(a = - i b \wedge a = i b\right) \vee \left(a = - i b \wedge b = 0\right) \vee \left(a = - i b \wedge a = i b \wedge b = 0\right) \vee a = - i b \\- e^{i b} \sin{\left(b \right)} - i e^{i b} \cos{\left(b \right)} + \frac{\sin^{2}{\left(b \right)}}{2} + \frac{\cos^{2}{\left(b \right)}}{2} - \frac{i e^{2 i b}}{2 b} + \frac{i e^{i b} \sin{\left(b \right)}}{b} - \frac{\sin{\left(b \right)} \cos{\left(b \right)}}{2 b} + \frac{i}{2 b} & \text{for}\: \left(a = i b \wedge b = 0\right) \vee a = i b \\\frac{e^{2 a}}{2 a} - \frac{1}{2 a} & \text{for}\: b = 0 \\\frac{a^{3} b \sin^{2}{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} + \frac{a^{3} b \cos^{2}{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} - \frac{a^{3} \sin{\left(b \right)} \cos{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} + \frac{a^{2} b e^{2 a}}{2 a^{3} b + 2 a b^{3}} - \frac{4 a^{2} b e^{a} \sin{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} - \frac{a^{2} b}{2 a^{3} b + 2 a b^{3}} + \frac{a b^{3} \sin^{2}{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} + \frac{a b^{3} \cos^{2}{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} + \frac{4 a b^{2} e^{a} \cos{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} - \frac{a b^{2} \sin{\left(b \right)} \cos{\left(b \right)}}{2 a^{3} b + 2 a b^{3}} - \frac{4 a b^{2}}{2 a^{3} b + 2 a b^{3}} + \frac{b^{3} e^{2 a}}{2 a^{3} b + 2 a b^{3}} - \frac{b^{3}}{2 a^{3} b + 2 a b^{3}} & \text{otherwise} \end{cases}$$
Piecewise((1, ((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))), (1 + cos(b)^2/2 + sin(b)^2/2 - 2/b + 2*cos(b)/b - cos(b)*sin(b)/(2*b), (a = 0)∨((a = 0)∧(a = i*b))∨((a = 0)∧(a = -i*b))∨((a = 0)∧(a = i*b)∧(a = -i*b))), (cos(b)^2/2 + sin(b)^2/2 - exp(-i*b)*sin(b) - i/(2*b) + i*cos(b)*exp(-i*b) + i*exp(-2*i*b)/(2*b) - cos(b)*sin(b)/(2*b) - i*exp(-i*b)*sin(b)/b, (a = -i*b)∨((b = 0)∧(a = -i*b))∨((a = i*b)∧(a = -i*b))∨((b = 0)∧(a = i*b)∧(a = -i*b))), (cos(b)^2/2 + sin(b)^2/2 + i/(2*b) - exp(i*b)*sin(b) - i*cos(b)*exp(i*b) - i*exp(2*i*b)/(2*b) - cos(b)*sin(b)/(2*b) + i*exp(i*b)*sin(b)/b, (a = i*b)∨((b = 0)∧(a = i*b))), (-1/(2*a) + exp(2*a)/(2*a), b = 0), (-b^3/(2*a*b^3 + 2*b*a^3) + b^3*exp(2*a)/(2*a*b^3 + 2*b*a^3) - b*a^2/(2*a*b^3 + 2*b*a^3) - 4*a*b^2/(2*a*b^3 + 2*b*a^3) + a*b^3*cos(b)^2/(2*a*b^3 + 2*b*a^3) + a*b^3*sin(b)^2/(2*a*b^3 + 2*b*a^3) + b*a^2*exp(2*a)/(2*a*b^3 + 2*b*a^3) + b*a^3*cos(b)^2/(2*a*b^3 + 2*b*a^3) + b*a^3*sin(b)^2/(2*a*b^3 + 2*b*a^3) - a^3*cos(b)*sin(b)/(2*a*b^3 + 2*b*a^3) - a*b^2*cos(b)*sin(b)/(2*a*b^3 + 2*b*a^3) - 4*b*a^2*exp(a)*sin(b)/(2*a*b^3 + 2*b*a^3) + 4*a*b^2*cos(b)*exp(a)/(2*a*b^3 + 2*b*a^3), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.