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Solving integrals/ -3ysinx+12ycosx

Integral of -3ysinx+12ycosx dx

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The solution

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 2*pi                              
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  |  (-3*y*sin(x) + 12*y*cos(x)) dx
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∫02π(−3ysin⁡(x)+12ycos⁡(x)) dx\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

  1. Integrate term-by-term:

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

      ∫−3ysin⁡(x) dx=−3y∫sin⁡(x) dx\int - 3 y \sin{\left(x \right)}\, dx = - 3 y \int \sin{\left(x \right)}\, dx

      1. The integral of sine is negative cosine:

        ∫sin⁡(x) dx=−cos⁡(x)\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}

      So, the result is: 3ycos⁡(x)3 y \cos{\left(x \right)}

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

      ∫12ycos⁡(x) dx=12y∫cos⁡(x) dx\int 12 y \cos{\left(x \right)}\, dx = 12 y \int \cos{\left(x \right)}\, dx

      1. The integral of cosine is sine:

        ∫cos⁡(x) dx=sin⁡(x)\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}

      So, the result is: 12ysin⁡(x)12 y \sin{\left(x \right)}

    The result is: 12ysin⁡(x)+3ycos⁡(x)12 y \sin{\left(x \right)} + 3 y \cos{\left(x \right)}

  2. Now simplify:

    3y(4sin⁡(x)+cos⁡(x))3 y \left(4 \sin{\left(x \right)} + \cos{\left(x \right)}\right)

  3. Add the constant of integration:

    3y(4sin⁡(x)+cos⁡(x))+constant3 y \left(4 \sin{\left(x \right)} + \cos{\left(x \right)}\right)+ \mathrm{constant}

The answer is:

3y(4sin⁡(x)+cos⁡(x))+constant3 y \left(4 \sin{\left(x \right)} + \cos{\left(x \right)}\right)+ \mathrm{constant}

The answer (Indefinite) [src] 
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 | (-3*y*sin(x) + 12*y*cos(x)) dx = C + 3*y*cos(x) + 12*y*sin(x)
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∫(−3ysin⁡(x)+12ycos⁡(x)) dx=C+12ysin⁡(x)+3ycos⁡(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] 
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    Use the examples entering the upper and lower limits of integration.

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