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Solving integrals/ 8*cos(4x-12)

Integral of 8*cos(4x-12) dx

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The solution

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 |  8*cos(4*x - 12) dx
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∫018cos⁡(4x−12) dx\int\limits_{0}^{1} 8 \cos{\left(4 x - 12 \right)}\, dx 
Integral(8*cos(4*x - 12), (x, 0, 1))
Detail solution

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

    ∫8cos⁡(4x−12) dx=8∫cos⁡(4x−12) dx\int 8 \cos{\left(4 x - 12 \right)}\, dx = 8 \int \cos{\left(4 x - 12 \right)}\, dx

    1. Let u=4x−12u = 4 x - 12.

      Then let du=4dxdu = 4 dx and substitute du4\frac{du}{4}:

      ∫cos⁡(u)4 du\int \frac{\cos{\left(u \right)}}{4}\, du

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

        ∫cos⁡(u) du=∫cos⁡(u) du4\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{4}

        1. The integral of cosine is sine:

          ∫cos⁡(u) du=sin⁡(u)\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}

        So, the result is: sin⁡(u)4\frac{\sin{\left(u \right)}}{4}

      Now substitute uu back in:

      sin⁡(4x−12)4\frac{\sin{\left(4 x - 12 \right)}}{4}

    So, the result is: 2sin⁡(4x−12)2 \sin{\left(4 x - 12 \right)}

  2. Now simplify:

    2sin⁡(4x−12)2 \sin{\left(4 x - 12 \right)}

  3. Add the constant of integration:

    2sin⁡(4x−12)+constant2 \sin{\left(4 x - 12 \right)}+ \mathrm{constant}

The answer is:

2sin⁡(4x−12)+constant2 \sin{\left(4 x - 12 \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
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 | 8*cos(4*x - 12) dx = C + 2*sin(4*x - 12)
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∫8cos⁡(4x−12) dx=C+2sin⁡(4x−12)\int 8 \cos{\left(4 x - 12 \right)}\, dx = C + 2 \sin{\left(4 x - 12 \right)}
Simplify
The graph
0.003.000.250.500.751.001.251.501.752.002.252.502.75-2020
Plot the graph f(x)
The answer [src] 
-2*sin(8) + 2*sin(12)
−2sin⁡(8)+2sin⁡(12)- 2 \sin{\left(8 \right)} + 2 \sin{\left(12 \right)} 
=
= 
-2*sin(8) + 2*sin(12)
−2sin⁡(8)+2sin⁡(12)- 2 \sin{\left(8 \right)} + 2 \sin{\left(12 \right)} 
-2*sin(8) + 2*sin(12)
Numerical answer [src] 
-3.05186232924763
-3.05186232924763

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

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