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- Integral of d{x}:
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Integral of (1+x)/x
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Integral of 1/√(1-x²)
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Integral of e^(x+1)
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Identical expressions
- e^(ax)sin(bx)
- e to the power of (ax) sinus of (bx)
- e(ax)sin(bx)
- eaxsinbx
- e^axsinbx
- e^(ax)sin(bx)dx
Integral of e^(ax)sin(bx) dx
The solution
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[src]
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)
[src]
// /cos(b*x) x*sin(b*x) \
|| |-------- + ---------- for b != 0 |
|| | 2 b |
|| | b |
|| < for a = 0|
|| | 2 |
|| | x |
|| | -- otherwise |
|| \ 2 |
|| |
||/ x for And(a = 0, b = 0) |
/ // x for a = 0\ ||| |
| || | ||| -I*b*x -I*b*x -I*b*x |
| a*x || a*x | |||x*cos(b*x)*e e *sin(b*x) I*x*e *sin(b*x) |
| e *sin(b*x) dx = C + |
$${{e^{a\,x}\,\left(a\,\sin \left(b\,x\right)-b\,\cos \left(b\,x
\right)\right)}\over{b^2+a^2}}$$
The answer
[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)) | | -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
$${{a\,e^{a}\,\sin b-e^{a}\,b\,\cos b}\over{b^2+a^2}}+{{b}\over{b^2+a
^2}}$$
=
=
/ 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
$$\begin{cases} 0 & \text{for}\: \left(a = 0 \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge a = i b \wedge b = 0\right) \\\frac{e^{- i b} \sin{\left(b \right)}}{2} - \frac{i e^{- i b} \cos{\left(b \right)}}{2} + \frac{1}{2 b} - \frac{e^{- i b} \cos{\left(b \right)}}{2 b} & \text{for}\: \left(a = 0 \wedge a = - i b\right) \vee \left(a = - i b \wedge a = i b\right) \vee \left(a = - i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge a = i b\right) \vee \left(a = - i b \wedge a = i b \wedge b = 0\right) \vee a = - i b \\\frac{e^{i b} \sin{\left(b \right)}}{2} + \frac{i e^{i b} \cos{\left(b \right)}}{2} - \frac{e^{i b} \cos{\left(b \right)}}{2 b} + \frac{1}{2 b} & \text{for}\: \left(a = 0 \wedge a = i b\right) \vee \left(a = i b \wedge b = 0\right) \vee a = i b \\\frac{a e^{a} \sin{\left(b \right)}}{a^{2} + b^{2}} - \frac{b e^{a} \cos{\left(b \right)}}{a^{2} + b^{2}} + \frac{b}{a^{2} + b^{2}} & \text{otherwise} \end{cases}$$
Use the examples entering the upper and lower limits of integration.