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2*cos(2*x)
Solving integrals/ 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=2∫cos⁡(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)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)2 du=∫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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sin⁡(2 x)\sin \left(2\,x\right)
Simplify
The graph
0.000.050.100.150.200.250.300.350.400.450.5004
Plot the graph f(x)
The answer [src] 
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32\frac{\sqrt{3}}{2} 
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32\frac{\sqrt{3}}{2}
Simplify
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.

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