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Identical expressions
one /(five -4cos(2x))
1 divide by (5 minus 4 co sinus of e of (2x))
one divide by (five minus 4 co sinus of e of (2x))
1/5-4cos2x
1 divide by (5-4cos(2x))
1/(5-4cos(2x))dx
Similar expressions
1/(5+4cos(2x))

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

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

The solution

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  1                    
  /                    
 |                     
 |          1          
 |  1*-------------- dx
 |    5 - 4*cos(2*x)   
 |                     
/                      
0

$$\int\limits_{0}^{1} 1 \cdot \frac{1}{- 4 \cos{\left(2 x \right)} + 5}\, dx$$

Integral(1/(5 - 4*cos(2*x)), (x, 0, 1))

Detail solution

Rewrite the integrand:

$1 \cdot \frac{1}{5 - 4 \cos{\left(2 x \right)}} = - \frac{1}{4 \cos{\left(2 x \right)} - 5}$
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{1}{4 \cos{\left(2 x \right)} - 5}\right)\, dx = - \int \frac{1}{4 \cos{\left(2 x \right)} - 5}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $- \frac{\operatorname{atan}{\left(3 \tan{\left(x \right)} \right)}}{3} - \frac{\pi \left\lfloor{\frac{x - \frac{\pi}{2}}{\pi}}\right\rfloor}{3}$
So, the result is: $\frac{\operatorname{atan}{\left(3 \tan{\left(x \right)} \right)}}{3} + \frac{\pi \left\lfloor{\frac{x - \frac{\pi}{2}}{\pi}}\right\rfloor}{3}$
Now simplify:

$\frac{\operatorname{atan}{\left(3 \tan{\left(x \right)} \right)}}{3} + \frac{\pi \left\lfloor{\frac{x}{\pi} - \frac{1}{2}}\right\rfloor}{3}$
Add the constant of integration:

$\frac{\operatorname{atan}{\left(3 \tan{\left(x \right)} \right)}}{3} + \frac{\pi \left\lfloor{\frac{x}{\pi} - \frac{1}{2}}\right\rfloor}{3}+ \mathrm{constant}$

The answer is:

$\frac{\operatorname{atan}{\left(3 \tan{\left(x \right)} \right)}}{3} + \frac{\pi \left\lfloor{\frac{x}{\pi} - \frac{1}{2}}\right\rfloor}{3}+ \mathrm{constant}$

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        
 |                                                            
/

$${{\arctan \left({{3\,\sin \left(2\,x\right)}\over{\cos \left(2\,x \right)+1}}\right)}\over{3}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

atan(3*tan(1))
--------------
      3

$${{\arctan \left({{3\,\sin 2}\over{\cos 2+1}}\right)}\over{3}}$$

atan(3*tan(1))
--------------
      3

$$\frac{\operatorname{atan}{\left(3 \tan{\left(1 \right)} \right)}}{3}$$

Simplify

Numerical answer [src]

0.453315553466098

0.453315553466098

The graph

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

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Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution