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Solving integrals/ exp(-ax)cos(bx)

Integral of exp(-ax)cos(bx) dx

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The solution

You have entered [src] 
  1                  
  /                  
 |                   
 |   -a*x            
 |  e    *cos(b*x) dx
 |                   
/                    
0
$$\int\limits_{0}^{1} e^{- a x} \cos{\left(b x \right)}\, dx$$ 
Integral(exp((-a)*x)*cos(b*x), (x, 0, 1))
The answer (Indefinite) [src] 
                             //                            /sin(b*x)   x*cos(b*x)                                                   \                                 
                             ||                            |-------- - ----------  for b != 0                                       |                                 
                             ||                            |    2          b                                                        |                                 
                             ||                            <   b                                                           for a = 0|                                 
                             ||                            |                                                                        |                                 
                             ||                            |          0            otherwise                                        |                                 
                             ||                            \                                                                        |                                 
                             ||                                                                                                     |                                 
                             || //                              0                                 for And(a = 0, b = 0)\            |                                 
                             || ||                                                                                     |            |                                 
  /                          || ||    I*b*x                          I*b*x      I*b*x                                  |            |   //   x     for a = 0\         
 |                           || || x*e     *sin(b*x)   I*x*cos(b*x)*e        I*e     *sin(b*x)                         |            |   ||                  |         
 |  -a*x                     || || ----------------- + ------------------- - -----------------        for a = -I*b     |            |   ||  -a*x            |         
 | e    *cos(b*x) dx = C + b*|< ||         2                    2                   2*b                                |            | + |<-e                |*cos(b*x)
 |                           || ||                                                                                     |            |   ||-------  otherwise|         
/                            || ||   -I*b*x               -I*b*x                          -I*b*x                       |            |   ||   a              |         
                             ||-|
$$\int e^{- a x} \cos{\left(b x \right)}\, dx = C + b \left(\begin{cases} \begin{cases} - \frac{x \cos{\left(b x \right)}}{b} + \frac{\sin{\left(b x \right)}}{b^{2}} & \text{for}\: b \neq 0 \\0 & \text{otherwise} \end{cases} & \text{for}\: a = 0 \\- \frac{\begin{cases} 0 & \text{for}\: a = 0 \wedge b = 0 \\\frac{x e^{i b x} \sin{\left(b x \right)}}{2} + \frac{i x e^{i b x} \cos{\left(b x \right)}}{2} - \frac{i e^{i b x} \sin{\left(b x \right)}}{2 b} & \text{for}\: a = - i b \\\frac{x e^{- i b x} \sin{\left(b x \right)}}{2} - \frac{i x e^{- i b x} \cos{\left(b x \right)}}{2} + \frac{i e^{- i b x} \sin{\left(b x \right)}}{2 b} & \text{for}\: a = i b \\- \frac{a \sin{\left(b x \right)}}{a^{2} e^{a x} + b^{2} e^{a x}} - \frac{b \cos{\left(b x \right)}}{a^{2} e^{a x} + b^{2} e^{a x}} & \text{otherwise} \end{cases}}{a} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} x & \text{for}\: a = 0 \\- \frac{e^{- a x}}{a} & \text{otherwise} \end{cases}\right) \cos{\left(b x \right)}$$
Simplify
The answer [src] 
/                     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))             
|                                                                                                                                                                                                
|         I*b    I*b             I*b                                                                                                                                                             
| cos(b)*e      e   *sin(b)   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*b             2                                                                                                                                                            
|                                                                                                                                                                                                
|        -I*b      -I*b           -I*b
$$\begin{cases} 1 & \text{for}\: \left(a = 0 \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge a = i b \wedge b = 0\right) \\- \frac{i e^{i b} \sin{\left(b \right)}}{2} + \frac{e^{i b} \cos{\left(b \right)}}{2} + \frac{e^{i b} \sin{\left(b \right)}}{2 b} & \text{for}\: \left(a = 0 \wedge a = - i b\right) \vee \left(a = - i b \wedge a = i b\right) \vee \left(a = - i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge a = i b\right) \vee \left(a = - i b \wedge a = i b \wedge b = 0\right) \vee a = - i b \\\frac{i e^{- i b} \sin{\left(b \right)}}{2} + \frac{e^{- i b} \cos{\left(b \right)}}{2} + \frac{e^{- i b} \sin{\left(b \right)}}{2 b} & \text{for}\: \left(a = 0 \wedge a = i b\right) \vee \left(a = i b \wedge b = 0\right) \vee a = i b \\- \frac{a \cos{\left(b \right)}}{a^{2} e^{a} + b^{2} e^{a}} + \frac{a}{a^{2} + b^{2}} + \frac{b \sin{\left(b \right)}}{a^{2} e^{a} + b^{2} e^{a}} & \text{otherwise} \end{cases}$$ 
=
= 
/                     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))             
|                                                                                                                                                                                                
|         I*b    I*b             I*b                                                                                                                                                             
| cos(b)*e      e   *sin(b)   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*b             2                                                                                                                                                            
|                                                                                                                                                                                                
|        -I*b      -I*b           -I*b
$$\begin{cases} 1 & \text{for}\: \left(a = 0 \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge a = i b \wedge b = 0\right) \\- \frac{i e^{i b} \sin{\left(b \right)}}{2} + \frac{e^{i b} \cos{\left(b \right)}}{2} + \frac{e^{i b} \sin{\left(b \right)}}{2 b} & \text{for}\: \left(a = 0 \wedge a = - i b\right) \vee \left(a = - i b \wedge a = i b\right) \vee \left(a = - i b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - i b \wedge a = i b\right) \vee \left(a = - i b \wedge a = i b \wedge b = 0\right) \vee a = - i b \\\frac{i e^{- i b} \sin{\left(b \right)}}{2} + \frac{e^{- i b} \cos{\left(b \right)}}{2} + \frac{e^{- i b} \sin{\left(b \right)}}{2 b} & \text{for}\: \left(a = 0 \wedge a = i b\right) \vee \left(a = i b \wedge b = 0\right) \vee a = i b \\- \frac{a \cos{\left(b \right)}}{a^{2} e^{a} + b^{2} e^{a}} + \frac{a}{a^{2} + b^{2}} + \frac{b \sin{\left(b \right)}}{a^{2} e^{a} + b^{2} e^{a}} & \text{otherwise} \end{cases}$$ 
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))), (cos(b)*exp(i*b)/2 + exp(i*b)*sin(b)/(2*b) - i*exp(i*b)*sin(b)/2, (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(b)*exp(-i*b)/2 + i*exp(-i*b)*sin(b)/2 + exp(-i*b)*sin(b)/(2*b), (a = i*b)∨((a = 0)∧(a = i*b))∨((b = 0)∧(a = i*b))), (a/(a^2 + b^2) + b*sin(b)/(a^2*exp(a) + b^2*exp(a)) - a*cos(b)/(a^2*exp(a) + b^2*exp(a)), True))

    Use the examples entering the upper and lower limits of integration.

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