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1/1+cos(6x)
Solving integrals/ 1/1+cos(6x)

Integral of 1/1+cos(6x) dx

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The solution

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 |  (1 + cos(6*x)) dx
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∫01(cos⁡(6x)+1) dx\int\limits_{0}^{1} \left(\cos{\left(6 x \right)} + 1\right)\, dx 
Integral(1 + cos(6*x), (x, 0, 1))
Detail solution

  1. Integrate term-by-term:

    1. Let u=6xu = 6 x.

      Then let du=6dxdu = 6 dx and substitute du6\frac{du}{6}:

      ∫cos⁡(u)36 du\int \frac{\cos{\left(u \right)}}{36}\, du

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

        ∫cos⁡(u)6 du=∫cos⁡(u) du6\int \frac{\cos{\left(u \right)}}{6}\, du = \frac{\int \cos{\left(u \right)}\, du}{6}

        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)6\frac{\sin{\left(u \right)}}{6}

      Now substitute uu back in:

      sin⁡(6x)6\frac{\sin{\left(6 x \right)}}{6}

    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⁡(6x)6x + \frac{\sin{\left(6 x \right)}}{6}

  2. Add the constant of integration:

    x+sin⁡(6x)6+constantx + \frac{\sin{\left(6 x \right)}}{6}+ \mathrm{constant}

The answer is:

x+sin⁡(6x)6+constantx + \frac{\sin{\left(6 x \right)}}{6}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                    
 |                             sin(6*x)
 | (1 + cos(6*x)) dx = C + x + --------
 |                                6    
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sin⁡(6 x)6+x{{\sin \left(6\,x\right)}\over{6}}+x
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9004
Plot the graph f(x)
The answer [src] 
    sin(6)
1 + ------
      6
sin⁡6+66{{\sin 6+6}\over{6}} 
=
= 
    sin(6)
1 + ------
      6
sin⁡(6)6+1\frac{\sin{\left(6 \right)}}{6} + 1
Simplify
Numerical answer [src] 
0.953430750300179
0.953430750300179
The graph
Integral of 1/1+cos(6x) dx

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

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