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1/(5-4cos(2x))
Solving integrals/ 1/(5-4cos(2x))

Integral of 1/(5-4cos(2x)) dx

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The solution

You have entered [src] 
  1                    
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 |          1          
 |  1*-------------- dx
 |    5 - 4*cos(2*x)   
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0
∫011⋅1−4cos⁡(2x)+5 dx\int\limits_{0}^{1} 1 \cdot \frac{1}{- 4 \cos{\left(2 x \right)} + 5}\, dx 
Integral(1/(5 - 4*cos(2*x)), (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    1⋅15−4cos⁡(2x)=−14cos⁡(2x)−51 \cdot \frac{1}{5 - 4 \cos{\left(2 x \right)}} = - \frac{1}{4 \cos{\left(2 x \right)} - 5}

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

    ∫(−14cos⁡(2x)−5) dx=−∫14cos⁡(2x)−5 dx\int \left(- \frac{1}{4 \cos{\left(2 x \right)} - 5}\right)\, dx = - \int \frac{1}{4 \cos{\left(2 x \right)} - 5}\, dx

    1. Don't know the steps in finding this integral.

      But the integral is

      −atan⁡(3tan⁡(x))3−π⌊x−π2π⌋3- \frac{\operatorname{atan}{\left(3 \tan{\left(x \right)} \right)}}{3} - \frac{\pi \left\lfloor{\frac{x - \frac{\pi}{2}}{\pi}}\right\rfloor}{3}

    So, the result is: atan⁡(3tan⁡(x))3+π⌊x−π2π⌋3\frac{\operatorname{atan}{\left(3 \tan{\left(x \right)} \right)}}{3} + \frac{\pi \left\lfloor{\frac{x - \frac{\pi}{2}}{\pi}}\right\rfloor}{3}

  3. Now simplify:

    atan⁡(3tan⁡(x))3+π⌊xπ−12⌋3\frac{\operatorname{atan}{\left(3 \tan{\left(x \right)} \right)}}{3} + \frac{\pi \left\lfloor{\frac{x}{\pi} - \frac{1}{2}}\right\rfloor}{3}

  4. Add the constant of integration:

    atan⁡(3tan⁡(x))3+π⌊xπ−12⌋3+constant\frac{\operatorname{atan}{\left(3 \tan{\left(x \right)} \right)}}{3} + \frac{\pi \left\lfloor{\frac{x}{\pi} - \frac{1}{2}}\right\rfloor}{3}+ \mathrm{constant}

The answer is:

atan⁡(3tan⁡(x))3+π⌊xπ−12⌋3+constant\frac{\operatorname{atan}{\left(3 \tan{\left(x \right)} \right)}}{3} + \frac{\pi \left\lfloor{\frac{x}{\pi} - \frac{1}{2}}\right\rfloor}{3}+ \mathrm{constant}

The answer (Indefinite) [src] 
                                                      /    pi\
                                                      |x - --|
  /                                                   |    2 |
 |                                            pi*floor|------|
 |         1                 atan(3*tan(x))           \  pi  /
 | 1*-------------- dx = C + -------------- + ----------------
 |   5 - 4*cos(2*x)                3                 3        
 |                                                            
/
arctan⁡(3 sin⁡(2 x)cos⁡(2 x)+1)3{{\arctan \left({{3\,\sin \left(2\,x\right)}\over{\cos \left(2\,x \right)+1}}\right)}\over{3}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.902-2
Plot the graph f(x)
The answer [src] 
atan(3*tan(1))
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      3
arctan⁡(3 sin⁡2cos⁡2+1)3{{\arctan \left({{3\,\sin 2}\over{\cos 2+1}}\right)}\over{3}} 
=
= 
atan(3*tan(1))
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      3
atan⁡(3tan⁡(1))3\frac{\operatorname{atan}{\left(3 \tan{\left(1 \right)} \right)}}{3}
Simplify
Numerical answer [src] 
0.453315553466098
0.453315553466098
The graph
Integral of 1/(5-4cos(2x)) dx

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

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