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2*cos(2*x)

Integral of 2*cos(2*x) dx

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The solution

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0π62cos(2x)dx\int\limits_{0}^{\frac{\pi}{6}} 2 \cos{\left(2 x \right)}\, dx
Integral(2*cos(2*x), (x, 0, pi/6))
Detail solution
  1. The integral of a constant times a function is the constant times the integral of the function:

    2cos(2x)dx=2cos(2x)dx\int 2 \cos{\left(2 x \right)}\, dx = 2 \int \cos{\left(2 x \right)}\, dx

    1. Let u=2xu = 2 x.

      Then let du=2dxdu = 2 dx and substitute du2\frac{du}{2}:

      cos(u)4du\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)2du=cos(u)du2\int \frac{\cos{\left(u \right)}}{2}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}

        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)2\frac{\sin{\left(u \right)}}{2}

      Now substitute uu back in:

      sin(2x)2\frac{\sin{\left(2 x \right)}}{2}

    So, the result is: sin(2x)\sin{\left(2 x \right)}

  2. Add the constant of integration:

    sin(2x)+constant\sin{\left(2 x \right)}+ \mathrm{constant}


The answer is:

sin(2x)+constant\sin{\left(2 x \right)}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                            
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 | 2*cos(2*x) dx = C + sin(2*x)
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sin(2x)\sin \left(2\,x\right)
The graph
0.000.050.100.150.200.250.300.350.400.450.5004
The answer [src]
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32\frac{\sqrt{3}}{2}
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32\frac{\sqrt{3}}{2}
Numerical answer [src]
0.866025403784439
0.866025403784439
The graph
Integral of 2*cos(2*x) dx

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