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Integral of d{x}:
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Identical expressions
-3ysinx+12ycosx
minus 3y sinus of x plus 12y co sinus of e of x
-3ysinx+12ycosxdx
Similar expressions
3ysinx+12ycosx
-3ysinx-12ycosx

Integral of -3ysinx+12ycosx dx

The solution

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 2*pi                              
   /                               
  |                                
  |  (-3*y*sin(x) + 12*y*cos(x)) dx
  |                                
 /                                 
 0

$$\int\limits_{0}^{2 \pi} \left(- 3 y \sin{\left(x \right)} + 12 y \cos{\left(x \right)}\right)\, dx$$

Integral((-3*y)*sin(x) + (12*y)*cos(x), (x, 0, 2*pi))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int - 3 y \sin{\left(x \right)}\, dx = - 3 y \int \sin{\left(x \right)}\, dx$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
  So, the result is: $3 y \cos{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 12 y \cos{\left(x \right)}\, dx = 12 y \int \cos{\left(x \right)}\, dx$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
  So, the result is: $12 y \sin{\left(x \right)}$
The result is: $12 y \sin{\left(x \right)} + 3 y \cos{\left(x \right)}$
Now simplify:

$3 y \left(4 \sin{\left(x \right)} + \cos{\left(x \right)}\right)$
Add the constant of integration:

$3 y \left(4 \sin{\left(x \right)} + \cos{\left(x \right)}\right)+ \mathrm{constant}$

The answer is:

$3 y \left(4 \sin{\left(x \right)} + \cos{\left(x \right)}\right)+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                             
 |                                                              
 | (-3*y*sin(x) + 12*y*cos(x)) dx = C + 3*y*cos(x) + 12*y*sin(x)
 |                                                              
/

$$\int \left(- 3 y \sin{\left(x \right)} + 12 y \cos{\left(x \right)}\right)\, dx = C + 12 y \sin{\left(x \right)} + 3 y \cos{\left(x \right)}$$

Simplify

The answer [src]

$$0$$

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

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Integral of -3ysinx+12ycosx dx

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The solution