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Solving integrals/ e^(x)*sin(n*x)

Integral of e^(x)*sin(n*x) dx

Limits of integration:

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The solution

You have entered [src] 
  x               
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 |   x            
 |  E *sin(n*x) dx
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0
$$\int\limits_{0}^{x} e^{x} \sin{\left(n x \right)}\, dx$$ 
Integral(E^x*sin(n*x), (x, 0, x))
The answer (Indefinite) [src] 
                        //   /         x      x                      x\            \
                        ||   |cosh(x)*e    x*e *sinh(x)   x*cosh(x)*e |            |
                        ||-I*|---------- + ------------ - ------------|  for n = -I|
                        ||   \    2             2              2      /            |
  /                     ||                                                         |
 |                      ||  /         x      x                      x\             |
 |  x                   ||  |cosh(x)*e    x*e *sinh(x)   x*cosh(x)*e |             |
 | E *sin(n*x) dx = C + |
$$\int e^{x} \sin{\left(n x \right)}\, dx = C + \begin{cases} - i \left(\frac{x e^{x} \sinh{\left(x \right)}}{2} - \frac{x e^{x} \cosh{\left(x \right)}}{2} + \frac{e^{x} \cosh{\left(x \right)}}{2}\right) & \text{for}\: n = - i \\i \left(\frac{x e^{x} \sinh{\left(x \right)}}{2} - \frac{x e^{x} \cosh{\left(x \right)}}{2} + \frac{e^{x} \cosh{\left(x \right)}}{2}\right) & \text{for}\: n = i \\- \frac{n e^{x} \cos{\left(n x \right)}}{n^{2} + 1} + \frac{e^{x} \sin{\left(n x \right)}}{n^{2} + 1} & \text{otherwise} \end{cases}$$
Simplify
The answer [src] 
          x                        x
  n      e *sin(n*x)   n*cos(n*x)*e 
------ + ----------- - -------------
     2           2              2   
1 + n       1 + n          1 + n
$$- \frac{n e^{x} \cos{\left(n x \right)}}{n^{2} + 1} + \frac{n}{n^{2} + 1} + \frac{e^{x} \sin{\left(n x \right)}}{n^{2} + 1}$$ 
=
= 
          x                        x
  n      e *sin(n*x)   n*cos(n*x)*e 
------ + ----------- - -------------
     2           2              2   
1 + n       1 + n          1 + n
$$- \frac{n e^{x} \cos{\left(n x \right)}}{n^{2} + 1} + \frac{n}{n^{2} + 1} + \frac{e^{x} \sin{\left(n x \right)}}{n^{2} + 1}$$ 
n/(1 + n^2) + exp(x)*sin(n*x)/(1 + n^2) - n*cos(n*x)*exp(x)/(1 + n^2)

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

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