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(1+cos2x)/2
Solving integrals/ (1+cos2x)/2

Integral of (1+cos2x)/2 dx

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The solution

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  1                
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 |  1 + cos(2*x)   
 |  ------------ dx
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∫01cos⁡(2x)+12 dx\int\limits_{0}^{1} \frac{\cos{\left(2 x \right)} + 1}{2}\, dx 
Integral((1 + cos(2*x))/2, (x, 0, 1))
Detail solution

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

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

    1. Integrate term-by-term:

      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}

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

        ∫1 dx=x\int 1\, dx = x

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

    So, the result is: x2+sin⁡(2x)4\frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4}

  2. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                                  
 |                                   
 | 1 + cos(2*x)          x   sin(2*x)
 | ------------ dx = C + - + --------
 |      2                2      4    
 |                                   
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sin⁡(2 x)2+x2{{{{\sin \left(2\,x\right)}\over{2}}+x}\over{2}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9002
Plot the graph f(x)
The answer [src] 
1   sin(2)
- + ------
2     4
sin⁡2+24{{\sin 2+2}\over{4}} 
=
= 
1   sin(2)
- + ------
2     4
sin⁡(2)4+12\frac{\sin{\left(2 \right)}}{4} + \frac{1}{2}
Упростить
Numerical answer [src] 
0.72732435670642
0.72732435670642
The graph
Integral of (1+cos2x)/2 dx

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

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