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Solving integrals/ 1/(5-3cosx)

Integral of 1/(5-3cosx) dx

The solution

You have entered [src]

  1                
  /                
 |                 
 |       1         
 |  ------------ dx
 |  5 - 3*cos(x)   
 |                 
/                  
0

$$\int\limits_{0}^{1} \frac{1}{5 - 3 \cos{\left(x \right)}}\, dx$$

Integral(1/(5 - 3*cos(x)), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\frac{1}{5 - 3 \cos{\left(x \right)}} = - \frac{1}{3 \cos{\left(x \right)} - 5}$
The integral of a constant times a function is the constant times the integral of the function:

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

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

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

The answer is:

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

The answer (Indefinite) [src]

                                                  /x   pi\
                                                  |- - --|
  /                          /     /x\\           |2   2 |
 |                       atan|2*tan|-||   pi*floor|------|
 |      1                    \     \2//           \  pi  /
 | ------------ dx = C + -------------- + ----------------
 | 5 - 3*cos(x)                2                 2        
 |                                                        
/

$$\int \frac{1}{5 - 3 \cos{\left(x \right)}}\, dx = C + \frac{\operatorname{atan}{\left(2 \tan{\left(\frac{x}{2} \right)} \right)}}{2} + \frac{\pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

atan(2*tan(1/2))
----------------
       2

$$\frac{\operatorname{atan}{\left(2 \tan{\left(\frac{1}{2} \right)} \right)}}{2}$$

atan(2*tan(1/2))
----------------
       2

$$\frac{\operatorname{atan}{\left(2 \tan{\left(\frac{1}{2} \right)} \right)}}{2}$$

atan(2*tan(1/2))/2

Numerical answer [src]

0.414811377137613

0.414811377137613

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/(5-3cosx) dx

Limits of integration:

The graph:

Piecewise:

The solution