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Solving integrals/ (e^x)*(x*cos(x)-y*sin(y))

Integral of (e^x)(xcos(x)-y*sin(y)) dx

The solution

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

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

Detail solution

Rewrite the integrand:

$e^{x} \left(x \cos{\left(x \right)} - y \sin{\left(y \right)}\right) = x e^{x} \cos{\left(x \right)} - y e^{x} \sin{\left(y \right)}$
Integrate term-by-term:
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = e^{x} \cos{\left(x \right)}$ .
  
  Then $\operatorname{du}{\left(x \right)} = 1$ .
  
  To find $v{\left(x \right)}$ :
  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,
      
      $\int e^{x} \cos{\left(x \right)}\, dx = \frac{e^{x} \sin{\left(x \right)}}{2} + \frac{e^{x} \cos{\left(x \right)}}{2}$
  Now evaluate the sub-integral.
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{e^{x} \sin{\left(x \right)}}{2}\, dx = \frac{\int e^{x} \sin{\left(x \right)}\, dx}{2}$
    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: $\frac{e^{x} \sin{\left(x \right)}}{4} - \frac{e^{x} \cos{\left(x \right)}}{4}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{e^{x} \cos{\left(x \right)}}{2}\, dx = \frac{\int e^{x} \cos{\left(x \right)}\, dx}{2}$
    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,
        
        $\int e^{x} \cos{\left(x \right)}\, dx = \frac{e^{x} \sin{\left(x \right)}}{2} + \frac{e^{x} \cos{\left(x \right)}}{2}$
    So, the result is: $\frac{e^{x} \sin{\left(x \right)}}{4} + \frac{e^{x} \cos{\left(x \right)}}{4}$
  The result is: $\frac{e^{x} \sin{\left(x \right)}}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- y e^{x} \sin{\left(y \right)}\right)\, dx = - y \sin{\left(y \right)} \int e^{x}\, dx$
  1. The integral of the exponential function is itself.
    
    $\int e^{x}\, dx = e^{x}$
  So, the result is: $- y e^{x} \sin{\left(y \right)}$
The result is: $x \left(\frac{e^{x} \sin{\left(x \right)}}{2} + \frac{e^{x} \cos{\left(x \right)}}{2}\right) - y e^{x} \sin{\left(y \right)} - \frac{e^{x} \sin{\left(x \right)}}{2}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

$$\int e^{x} \left(x \cos{\left(x \right)} - y \sin{\left(y \right)}\right)\, dx = C + x \left(\frac{e^{x} \sin{\left(x \right)}}{2} + \frac{e^{x} \cos{\left(x \right)}}{2}\right) - y e^{x} \sin{\left(y \right)} - \frac{e^{x} \sin{\left(x \right)}}{2}$$

Simplify

The answer [src]

           E*cos(1)             
y*sin(y) + -------- - E*y*sin(y)
              2

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

           E*cos(1)             
y*sin(y) + -------- - E*y*sin(y)
              2

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

y*sin(y) + E*cos(1)/2 - E*y*sin(y)

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (e^x)(xcos(x)-y*sin(y)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (e^x)*(x*cos(x)-y*sin(y)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Integral of (e^x)(xcos(x)-y*sin(y)) dx