Господин Экзамен
Интеграл exp(a*x)*sin(b*x) d{x}
Решение
Вы ввели
[src]
1 / | | a*x | e *sin(b*x) dx | / 0
$$\int\limits_{0}^{1} e^{a x} \sin{\left(b x \right)}\, dx$$
Ответ (Неопределённый)
[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 I*x*e *sin(b*x) I*cos(b*x)*e |
| e *sin(b*x) dx = C + |
$${{e^{a\,x}\,\left(a\,\sin \left(b\,x\right)-b\,\cos \left(b\,x
\right)\right)}\over{b^2+a^2}}$$
Ответ
[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 |e *sin(b) I*cos(b)*e I*e *sin(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 2 2*b | | I*b I*b I*b < e *sin(b) I*cos(b)*e I*e *sin(b) | ----------- + ------------- - ------------- for Or(And(a = 0, a = I*b), And(a = I*b, b = 0), a = I*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 |e *sin(b) I*cos(b)*e I*e *sin(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 2 2*b | | I*b I*b I*b < e *sin(b) I*cos(b)*e I*e *sin(b) | ----------- + ------------- - ------------- for Or(And(a = 0, a = I*b), And(a = I*b, b = 0), a = I*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
$$\begin{cases} 0 & \text{for}\: \left(a = 0 \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge a = i b \wedge b = 0\right) \\\frac{e^{- i b} \sin{\left(b \right)}}{2} - \frac{i e^{- i b} \cos{\left(b \right)}}{2} + \frac{i e^{- i b} \sin{\left(b \right)}}{2 b} & \text{for}\: \left(a = 0 \wedge a = - i b\right) \vee \left(a = - i b \wedge a = i b\right) \vee \left(a = - i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge a = i b\right) \vee \left(a = - i b \wedge a = i b \wedge b = 0\right) \vee a = - i b \\\frac{e^{i b} \sin{\left(b \right)}}{2} + \frac{i e^{i b} \cos{\left(b \right)}}{2} - \frac{i e^{i b} \sin{\left(b \right)}}{2 b} & \text{for}\: \left(a = 0 \wedge a = i b\right) \vee \left(a = i b \wedge b = 0\right) \vee a = i b \\\frac{a e^{a} \sin{\left(b \right)}}{a^{2} + b^{2}} - \frac{b e^{a} \cos{\left(b \right)}}{a^{2} + b^{2}} + \frac{b}{a^{2} + b^{2}} & \text{otherwise} \end{cases}$$
Данные примеры также можно применять при вводе верхнего и нижнего предела интегрирования.