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Solving integrals/ (cos(x))/(5-4*cos(x))

Integral of (cos(x))/(5-4*cos(x)) dx

The solution

You have entered [src]

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

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

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

Detail solution

Rewrite the integrand:

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

$\int \left(- \frac{\cos{\left(x \right)}}{4 \cos{\left(x \right)} - 5}\right)\, dx = - \int \frac{\cos{\left(x \right)}}{4 \cos{\left(x \right)} - 5}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\frac{x}{4} - \frac{5 \operatorname{atan}{\left(3 \tan{\left(\frac{x}{2} \right)} \right)}}{6} - \frac{5 \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor}{6}$
So, the result is: $- \frac{x}{4} + \frac{5 \operatorname{atan}{\left(3 \tan{\left(\frac{x}{2} \right)} \right)}}{6} + \frac{5 \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor}{6}$
Now simplify:

$- \frac{x}{4} + \frac{5 \operatorname{atan}{\left(3 \tan{\left(\frac{x}{2} \right)} \right)}}{6} + \frac{5 \pi \left\lfloor{\frac{x}{2 \pi} - \frac{1}{2}}\right\rfloor}{6}$
Add the constant of integration:

$- \frac{x}{4} + \frac{5 \operatorname{atan}{\left(3 \tan{\left(\frac{x}{2} \right)} \right)}}{6} + \frac{5 \pi \left\lfloor{\frac{x}{2 \pi} - \frac{1}{2}}\right\rfloor}{6}+ \mathrm{constant}$

The answer is:

$- \frac{x}{4} + \frac{5 \operatorname{atan}{\left(3 \tan{\left(\frac{x}{2} \right)} \right)}}{6} + \frac{5 \pi \left\lfloor{\frac{x}{2 \pi} - \frac{1}{2}}\right\rfloor}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

                                                          /x   pi\
                                                          |- - --|
  /                                /     /x\\             |2   2 |
 |                           5*atan|3*tan|-||   5*pi*floor|------|
 |    cos(x)             x         \     \2//             \  pi  /
 | ------------ dx = C - - + ---------------- + ------------------
 | 5 - 4*cos(x)          4          6                   6         
 |                                                                
/

$$\int \frac{\cos{\left(x \right)}}{5 - 4 \cos{\left(x \right)}}\, dx = C - \frac{x}{4} + \frac{5 \operatorname{atan}{\left(3 \tan{\left(\frac{x}{2} \right)} \right)}}{6} + \frac{5 \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor}{6}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  1   5*atan(3*tan(1/2))
- - + ------------------
  4           6

$$- \frac{1}{4} + \frac{5 \operatorname{atan}{\left(3 \tan{\left(\frac{1}{2} \right)} \right)}}{6}$$

  1   5*atan(3*tan(1/2))
- - + ------------------
  4           6

$$- \frac{1}{4} + \frac{5 \operatorname{atan}{\left(3 \tan{\left(\frac{1}{2} \right)} \right)}}{6}$$

-1/4 + 5*atan(3*tan(1/2))/6

Numerical answer [src]

0.60244819681611

0.60244819681611

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

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Identical expressions

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Integral of (cos(x))/(5-4*cos(x)) dx

Limits of integration:

The graph:

Piecewise:

The solution