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sin(2x)^2
Solving integrals/ sin(2x)^2

Integral of sin(2x)^2 dx

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The solution

You have entered [src] 
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∫01sin⁡2(2x) dx\int\limits_{0}^{1} \sin^{2}{\left(2 x \right)}\, dx 
Integral(sin(2*x)^2, (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    sin⁡2(2x)=12−cos⁡(4x)2\sin^{2}{\left(2 x \right)} = \frac{1}{2} - \frac{\cos{\left(4 x \right)}}{2}

  2. Integrate term-by-term:

    1. The integral of a constant is the constant times the variable of integration:

      ∫12 dx=x2\int \frac{1}{2}\, dx = \frac{x}{2}

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

      ∫(−cos⁡(4x)2) dx=−∫cos⁡(4x) dx2\int \left(- \frac{\cos{\left(4 x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(4 x \right)}\, dx}{2}

      1. Let u=4xu = 4 x.

        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)4\frac{\sin{\left(4 x \right)}}{4}

      So, the result is: −sin⁡(4x)8- \frac{\sin{\left(4 x \right)}}{8}

    The result is: x2−sin⁡(4x)8\frac{x}{2} - \frac{\sin{\left(4 x \right)}}{8}

  3. Add the constant of integration:

    x2−sin⁡(4x)8+constant\frac{x}{2} - \frac{\sin{\left(4 x \right)}}{8}+ \mathrm{constant}

The answer is:

x2−sin⁡(4x)8+constant\frac{x}{2} - \frac{\sin{\left(4 x \right)}}{8}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                               
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 |    2               x   sin(4*x)
 | sin (2*x) dx = C + - - --------
 |                    2      8    
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∫sin⁡2(2x) dx=C+x2−sin⁡(4x)8\int \sin^{2}{\left(2 x \right)}\, dx = C + \frac{x}{2} - \frac{\sin{\left(4 x \right)}}{8}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9002
Plot the graph f(x)
The answer [src] 
1   cos(2)*sin(2)
- - -------------
2         4
−sin⁡(2)cos⁡(2)4+12- \frac{\sin{\left(2 \right)} \cos{\left(2 \right)}}{4} + \frac{1}{2} 
=
= 
1   cos(2)*sin(2)
- - -------------
2         4
−sin⁡(2)cos⁡(2)4+12- \frac{\sin{\left(2 \right)} \cos{\left(2 \right)}}{4} + \frac{1}{2} 
1/2 - cos(2)*sin(2)/4
Numerical answer [src] 
0.594600311913491
0.594600311913491
The graph
Integral of sin(2x)^2 dx

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

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