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sin(4*x)/sin(2*x)
Solving integrals/ sin(4*x)/sin(2*x)

Integral of sin(4*x)/sin(2*x) dx

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The solution

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 |  sin(4*x)   
 |  -------- dx
 |  sin(2*x)   
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∫01sin⁡(4x)sin⁡(2x) dx\int\limits_{0}^{1} \frac{\sin{\left(4 x \right)}}{\sin{\left(2 x \right)}}\, dx 
Integral(sin(4*x)/sin(2*x), (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

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

  2. Integrate term-by-term:

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

      ∫2 dx=2x\int 2\, dx = 2 x

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

      ∫(−4sin⁡2(x)) dx=−4∫sin⁡2(x) dx\int \left(- 4 \sin^{2}{\left(x \right)}\right)\, dx = - 4 \int \sin^{2}{\left(x \right)}\, dx

      1. Rewrite the integrand:

        sin⁡2(x)=12−cos⁡(2x)2\sin^{2}{\left(x \right)} = \frac{1}{2} - \frac{\cos{\left(2 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⁡(2x)2) dx=−∫cos⁡(2x) dx2\int \left(- \frac{\cos{\left(2 x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(2 x \right)}\, dx}{2}

          1. Let u=2xu = 2 x.

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

            ∫cos⁡(u)2 du\int \frac{\cos{\left(u \right)}}{2}\, du

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

              ∫cos⁡(u) du=∫cos⁡(u) du2\int \cos{\left(u \right)}\, 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)4- \frac{\sin{\left(2 x \right)}}{4}

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

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

    The result is: sin⁡(2x)\sin{\left(2 x \right)}

  3. 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(4*x)                  
 | -------- dx = C + sin(2*x)
 | sin(2*x)                  
 |                           
/
∫sin⁡(4x)sin⁡(2x) dx=C+sin⁡(2x)\int \frac{\sin{\left(4 x \right)}}{\sin{\left(2 x \right)}}\, dx = C + \sin{\left(2 x \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.905-5
Plot the graph f(x)
The answer [src] 
sin(2)
sin⁡(2)\sin{\left(2 \right)} 
=
= 
sin(2)
sin⁡(2)\sin{\left(2 \right)} 
sin(2)
Numerical answer [src] 
0.909297426825682
0.909297426825682
The graph
Integral of sin(4*x)/sin(2*x) dx

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

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