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1+cos2x
Solving integrals/ 1+cos2x

Integral of 1+cos2x dx

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The solution

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

  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)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}

    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}

  2. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                                    
 |                             sin(2*x)
 | (1 + cos(2*x)) dx = C + x + --------
 |                                2    
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∫(cos⁡(2x)+1) dx=C+x+sin⁡(2x)2\int \left(\cos{\left(2 x \right)} + 1\right)\, dx = C + x + \frac{\sin{\left(2 x \right)}}{2}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9004
Plot the graph f(x)
The answer [src] 
    sin(2)
1 + ------
      2
sin⁡(2)2+1\frac{\sin{\left(2 \right)}}{2} + 1 
=
= 
    sin(2)
1 + ------
      2
sin⁡(2)2+1\frac{\sin{\left(2 \right)}}{2} + 1 
1 + sin(2)/2
Numerical answer [src] 
1.45464871341284
1.45464871341284
The graph
Integral of 1+cos2x dx

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

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