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Integral of x/x
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Integral of x*dx/(x+1)
- Integral of e^(a*x)*sin(b*x)
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Identical expressions
- e^(a*x)*sin(b*x)
- e to the power of (a multiply by x) multiply by sinus of (b multiply by x)
- e(a*x)*sin(b*x)
- ea*x*sinb*x
- e^(ax)sin(bx)
- e(ax)sin(bx)
- eaxsinbx
- e^axsinbx
- e^(a*x)*sin(b*x)dx
Integral of e^(a*x)*sin(b*x) dx
The solution
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1 / | | a*x | E *sin(b*x) dx | / 0
$$\int\limits_{0}^{1} e^{a x} \sin{\left(b x \right)}\, dx$$
Integral(E^(a*x)*sin(b*x), (x, 0, 1))
The answer (Indefinite)
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// 0 for And(a = 0, b = 0)\
|| |
|| -I*b*x -I*b*x -I*b*x |
||x*e *sin(b*x) cos(b*x)*e I*x*cos(b*x)*e |
||------------------ - ---------------- - -------------------- for a = -I*b |
/ || 2 2*b 2 |
| || |
| a*x || I*b*x I*b*x I*b*x |
| E *sin(b*x) dx = C + |< x*e *sin(b*x) cos(b*x)*e I*x*cos(b*x)*e |
| || ----------------- - --------------- + ------------------- for a = I*b |
/ || 2 2*b 2 |
|| |
|| a*x a*x |
|| a*e *sin(b*x) b*cos(b*x)*e |
|| --------------- - --------------- otherwise |
|| 2 2 2 2 |
\\ a + b a + b /
$$\int e^{a x} \sin{\left(b x \right)}\, dx = C + \begin{cases} 0 & \text{for}\: a = 0 \wedge b = 0 \\\frac{x e^{- i b x} \sin{\left(b x \right)}}{2} - \frac{i x e^{- i b x} \cos{\left(b x \right)}}{2} - \frac{e^{- i b x} \cos{\left(b x \right)}}{2 b} & \text{for}\: a = - i b \\\frac{x e^{i b x} \sin{\left(b x \right)}}{2} + \frac{i x e^{i b x} \cos{\left(b x \right)}}{2} - \frac{e^{i b x} \cos{\left(b x \right)}}{2 b} & \text{for}\: a = i b \\\frac{a e^{a x} \sin{\left(b x \right)}}{a^{2} + b^{2}} - \frac{b e^{a x} \cos{\left(b x \right)}}{a^{2} + b^{2}} & \text{otherwise} \end{cases}$$
The answer
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/ 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)) | | -I*b -I*b -I*b | 1 e *sin(b) I*cos(b)*e cos(b)*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 < 1 e *sin(b) I*cos(b)*e cos(b)*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 a*e *sin(b) b*cos(b)*e | ------- + ----------- - ----------- otherwise | 2 2 2 2 2 2 \ a + b a + b a + b
$$\begin{cases} 0 & \text{for}\: \left(a = 0 \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge a = i b \wedge b = 0\right) \\\frac{e^{- i b} \sin{\left(b \right)}}{2} - \frac{i e^{- i b} \cos{\left(b \right)}}{2} + \frac{1}{2 b} - \frac{e^{- i b} \cos{\left(b \right)}}{2 b} & \text{for}\: \left(a = 0 \wedge a = - i b\right) \vee \left(a = - i b \wedge a = i b\right) \vee \left(a = - i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge a = i b\right) \vee \left(a = - i b \wedge a = i b \wedge b = 0\right) \vee a = - i b \\\frac{e^{i b} \sin{\left(b \right)}}{2} + \frac{i e^{i b} \cos{\left(b \right)}}{2} - \frac{e^{i b} \cos{\left(b \right)}}{2 b} + \frac{1}{2 b} & \text{for}\: \left(a = 0 \wedge a = i b\right) \vee \left(a = i b \wedge b = 0\right) \vee a = i b \\\frac{a e^{a} \sin{\left(b \right)}}{a^{2} + b^{2}} - \frac{b e^{a} \cos{\left(b \right)}}{a^{2} + b^{2}} + \frac{b}{a^{2} + b^{2}} & \text{otherwise} \end{cases}$$
=
=
/ 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)) | | -I*b -I*b -I*b | 1 e *sin(b) I*cos(b)*e cos(b)*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 < 1 e *sin(b) I*cos(b)*e cos(b)*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 a*e *sin(b) b*cos(b)*e | ------- + ----------- - ----------- otherwise | 2 2 2 2 2 2 \ a + b a + b a + b
$$\begin{cases} 0 & \text{for}\: \left(a = 0 \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge a = i b \wedge b = 0\right) \\\frac{e^{- i b} \sin{\left(b \right)}}{2} - \frac{i e^{- i b} \cos{\left(b \right)}}{2} + \frac{1}{2 b} - \frac{e^{- i b} \cos{\left(b \right)}}{2 b} & \text{for}\: \left(a = 0 \wedge a = - i b\right) \vee \left(a = - i b \wedge a = i b\right) \vee \left(a = - i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge a = i b\right) \vee \left(a = - i b \wedge a = i b \wedge b = 0\right) \vee a = - i b \\\frac{e^{i b} \sin{\left(b \right)}}{2} + \frac{i e^{i b} \cos{\left(b \right)}}{2} - \frac{e^{i b} \cos{\left(b \right)}}{2 b} + \frac{1}{2 b} & \text{for}\: \left(a = 0 \wedge a = i b\right) \vee \left(a = i b \wedge b = 0\right) \vee a = i b \\\frac{a e^{a} \sin{\left(b \right)}}{a^{2} + b^{2}} - \frac{b e^{a} \cos{\left(b \right)}}{a^{2} + b^{2}} + \frac{b}{a^{2} + b^{2}} & \text{otherwise} \end{cases}$$
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))), (1/(2*b) + exp(-i*b)*sin(b)/2 - i*cos(b)*exp(-i*b)/2 - cos(b)*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))), (1/(2*b) + exp(i*b)*sin(b)/2 + i*cos(b)*exp(i*b)/2 - cos(b)*exp(i*b)/(2*b), (a = i*b)∨((a = 0)∧(a = i*b))∨((b = 0)∧(a = i*b))), (b/(a^2 + b^2) + a*exp(a)*sin(b)/(a^2 + b^2) - b*cos(b)*exp(a)/(a^2 + b^2), True))
Use the examples entering the upper and lower limits of integration.