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(cos(x)-xsin(x))y
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Integral of (cos(x)-xsin(x))y dx

The solution

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  x                         
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 |  (cos(x) - x*sin(x))*y dx
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x0

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

Integral((cos(x) - x*sin(x))*y, (x, x0, x))

Detail solution

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

$\int y \left(- x \sin{\left(x \right)} + \cos{\left(x \right)}\right)\, dx = y \int \left(- x \sin{\left(x \right)} + \cos{\left(x \right)}\right)\, dx$
1. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- x \sin{\left(x \right)}\right)\, dx = - \int x \sin{\left(x \right)}\, dx$
    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 $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$ .
      
      Then $\operatorname{du}{\left(x \right)} = 1$ .
      
      To find $v{\left(x \right)}$ :
      1. The integral of sine is negative cosine:
        
        $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
      Now evaluate the sub-integral.
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \cos{\left(x \right)}\right)\, dx = - \int \cos{\left(x \right)}\, dx$
      1. The integral of cosine is sine:
        
        $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
      So, the result is: $- \sin{\left(x \right)}$
    So, the result is: $x \cos{\left(x \right)} - \sin{\left(x \right)}$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
  The result is: $x \cos{\left(x \right)}$
So, the result is: $x y \cos{\left(x \right)}$
Add the constant of integration:

$x y \cos{\left(x \right)}+ \mathrm{constant}$

The answer is:

$x y \cos{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                         
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 | (cos(x) - x*sin(x))*y dx = C + x*y*cos(x)
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/

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

Simplify

The answer [src]

x*y*cos(x) - x0*y*cos(x0)

$$x y \cos{\left(x \right)} - x_{0} y \cos{\left(x_{0} \right)}$$

x*y*cos(x) - x0*y*cos(x0)

$$x y \cos{\left(x \right)} - x_{0} y \cos{\left(x_{0} \right)}$$

x*y*cos(x) - x0*y*cos(x0)

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

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Integral of (cos(x)-xsin(x))y dx

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Piecewise:

The solution