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Identical expressions
e^x(cosx+isinx)(one -i)
e to the power of x( co sinus of e of x plus i sinus of x)(1 minus i)
e to the power of x( co sinus of e of x plus i sinus of x)(one minus i)
ex(cosx+isinx)(1-i)
excosx+isinx1-i
e^xcosx+isinx1-i
e^x(cosx+isinx)(1-i)dx
Similar expressions
e^x(cosx+isinx)(1+i)
e^x(cosx-isinx)(1-i)

Solving integrals/ e^x(cosx+isinx)(1-i)

Integral of e^x(cosx+isinx)(1-i) dx

The solution

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 |  e *(cos(x) + I*sin(x))*(1 - I) dx
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0

$$\int\limits_{0}^{1} \left(1 - i\right) \left(i \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}\, dx$$

Integral(E^x*(cos(x) + i*sin(x))*(1 - i), (x, 0, 1))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int \left(1 - i\right) \left(i \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}\, dx = \left(1 - i\right) \int \left(i \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}\, dx$
1. Rewrite the integrand:
  
  $\left(i \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} = i e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int i e^{x} \sin{\left(x \right)}\, dx = i \int e^{x} \sin{\left(x \right)}\, dx$
    1. Use integration by parts, noting that the integrand eventually repeats itself.
      1. For the integrand $e^{x} \sin{\left(x \right)}$ :
        
        Let $u{\left(x \right)} = \sin{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{x}$ .
        
        Then $\int e^{x} \sin{\left(x \right)}\, dx = e^{x} \sin{\left(x \right)} - \int e^{x} \cos{\left(x \right)}\, dx$ .
      2. For the integrand $e^{x} \cos{\left(x \right)}$ :
        
        Let $u{\left(x \right)} = \cos{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{x}$ .
        
        Then $\int e^{x} \sin{\left(x \right)}\, dx = e^{x} \sin{\left(x \right)} - e^{x} \cos{\left(x \right)} + \int \left(- e^{x} \sin{\left(x \right)}\right)\, dx$ .
      3. Notice that the integrand has repeated itself, so move it to one side:
        
        $2 \int e^{x} \sin{\left(x \right)}\, dx = e^{x} \sin{\left(x \right)} - e^{x} \cos{\left(x \right)}$
        
        Therefore,
        
        $\int e^{x} \sin{\left(x \right)}\, dx = \frac{e^{x} \sin{\left(x \right)}}{2} - \frac{e^{x} \cos{\left(x \right)}}{2}$
    So, the result is: $i \left(\frac{e^{x} \sin{\left(x \right)}}{2} - \frac{e^{x} \cos{\left(x \right)}}{2}\right)$
  1. Use integration by parts, noting that the integrand eventually repeats itself.
    1. For the integrand $e^{x} \cos{\left(x \right)}$ :
      
      Let $u{\left(x \right)} = \cos{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{x}$ .
      
      Then $\int e^{x} \cos{\left(x \right)}\, dx = e^{x} \cos{\left(x \right)} - \int \left(- e^{x} \sin{\left(x \right)}\right)\, dx$ .
    2. For the integrand $- e^{x} \sin{\left(x \right)}$ :
      
      Let $u{\left(x \right)} = - \sin{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{x}$ .
      
      Then $\int e^{x} \cos{\left(x \right)}\, dx = e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)} + \int \left(- e^{x} \cos{\left(x \right)}\right)\, dx$ .
    3. Notice that the integrand has repeated itself, so move it to one side:
      
      $2 \int e^{x} \cos{\left(x \right)}\, dx = e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)}$
      
      Therefore,
      
      $\int e^{x} \cos{\left(x \right)}\, dx = \frac{e^{x} \sin{\left(x \right)}}{2} + \frac{e^{x} \cos{\left(x \right)}}{2}$
  The result is: $i \left(\frac{e^{x} \sin{\left(x \right)}}{2} - \frac{e^{x} \cos{\left(x \right)}}{2}\right) + \frac{e^{x} \sin{\left(x \right)}}{2} + \frac{e^{x} \cos{\left(x \right)}}{2}$
So, the result is: $\left(1 - i\right) \left(i \left(\frac{e^{x} \sin{\left(x \right)}}{2} - \frac{e^{x} \cos{\left(x \right)}}{2}\right) + \frac{e^{x} \sin{\left(x \right)}}{2} + \frac{e^{x} \cos{\left(x \right)}}{2}\right)$
Now simplify:

$\frac{\left(1 - i\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)} - \sqrt{2} i \cos{\left(x + \frac{\pi}{4} \right)}\right) e^{x}}{2}$
Add the constant of integration:

$\frac{\left(1 - i\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)} - \sqrt{2} i \cos{\left(x + \frac{\pi}{4} \right)}\right) e^{x}}{2}+ \mathrm{constant}$

The answer is:

$\frac{\left(1 - i\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)} - \sqrt{2} i \cos{\left(x + \frac{\pi}{4} \right)}\right) e^{x}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                   
 |                                                 /  / x                  x\           x    x       \
 |  x                                              |  |e *sin(x)   cos(x)*e |   cos(x)*e    e *sin(x)|
 | e *(cos(x) + I*sin(x))*(1 - I) dx = C + (1 - I)*|I*|--------- - ---------| + --------- + ---------|
 |                                                 \  \    2           2    /       2           2    /
/

$$\left(1-i\right)\,\left({{e^{x}\,\left(\sin x+\cos x\right)}\over{2 }}+{{i\,e^{x}\,\left(\sin x-\cos x\right)}\over{2}}\right)$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

        /e*cos(1)   e*sin(1)   e*I*sin(1)   e*I*cos(1)\           /1   I\
(1 - I)*|-------- + -------- + ---------- - ----------| - (1 - I)*|- - -|
        \   2          2           2            2     /           \2   2/

$$\left(1-i\right)\,\left({{\left(e\,i+e\right)\,\sin 1+\left(e-e\,i \right)\,\cos 1}\over{2}}+{{i-1}\over{2}}\right)$$

        /e*cos(1)   e*sin(1)   e*I*sin(1)   e*I*cos(1)\           /1   I\
(1 - I)*|-------- + -------- + ---------- - ----------| - (1 - I)*|- - -|
        \   2          2           2            2     /           \2   2/

$$\left(1 - i\right) \left(\frac{e \cos{\left(1 \right)}}{2} + \frac{e \sin{\left(1 \right)}}{2} - \frac{e i \cos{\left(1 \right)}}{2} + \frac{e i \sin{\left(1 \right)}}{2}\right) - \left(\frac{1}{2} - \frac{i}{2}\right) \left(1 - i\right)$$

Упростить

Numerical answer [src]

(2.28735528717884 - 0.468693939915885j)

(2.28735528717884 - 0.468693939915885j)

The graph

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

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Integral of e^x(cosx+isinx)(1-i) dx

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The solution