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Integral of 1/(4-3cos(x)) dx

The solution

You have entered [src]

 pi                
 --                
 2                 
  /                
 |                 
 |       1         
 |  ------------ dx
 |  4 - 3*cos(x)   
 |                 
/                  
0

$$\int\limits_{0}^{\frac{\pi}{2}} \frac{1}{4 - 3 \cos{\left(x \right)}}\, dx$$

Integral(1/(4 - 3*cos(x)), (x, 0, pi/2))

Detail solution

Rewrite the integrand:

$\frac{1}{4 - 3 \cos{\left(x \right)}} = - \frac{1}{3 \cos{\left(x \right)} - 4}$
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{1}{3 \cos{\left(x \right)} - 4}\right)\, dx = - \int \frac{1}{3 \cos{\left(x \right)} - 4}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $- \frac{2 \sqrt{7} \left(\operatorname{atan}{\left(\sqrt{7} \tan{\left(\frac{x}{2} \right)} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{7}$
So, the result is: $\frac{2 \sqrt{7} \left(\operatorname{atan}{\left(\sqrt{7} \tan{\left(\frac{x}{2} \right)} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{7}$
Now simplify:

$\frac{2 \sqrt{7} \left(\operatorname{atan}{\left(\sqrt{7} \tan{\left(\frac{x}{2} \right)} \right)} + \pi \left\lfloor{\frac{x}{2 \pi} - \frac{1}{2}}\right\rfloor\right)}{7}$
Add the constant of integration:

$\frac{2 \sqrt{7} \left(\operatorname{atan}{\left(\sqrt{7} \tan{\left(\frac{x}{2} \right)} \right)} + \pi \left\lfloor{\frac{x}{2 \pi} - \frac{1}{2}}\right\rfloor\right)}{7}+ \mathrm{constant}$

The answer is:

$\frac{2 \sqrt{7} \left(\operatorname{atan}{\left(\sqrt{7} \tan{\left(\frac{x}{2} \right)} \right)} + \pi \left\lfloor{\frac{x}{2 \pi} - \frac{1}{2}}\right\rfloor\right)}{7}+ \mathrm{constant}$

The answer (Indefinite) [src]

                                 /        /x   pi\                     \
                                 |        |- - --|                     |
  /                          ___ |        |2   2 |       /  ___    /x\\|
 |                       2*\/ 7 *|pi*floor|------| + atan|\/ 7 *tan|-|||
 |      1                        \        \  pi  /       \         \2///
 | ------------ dx = C + -----------------------------------------------
 | 4 - 3*cos(x)                                 7                       
 |                                                                      
/

$$\int \frac{1}{4 - 3 \cos{\left(x \right)}}\, dx = C + \frac{2 \sqrt{7} \left(\operatorname{atan}{\left(\sqrt{7} \tan{\left(\frac{x}{2} \right)} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{7}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       ___       ___ /          /  ___\\
2*pi*\/ 7    2*\/ 7 *\-pi + atan\\/ 7 //
---------- + ---------------------------
    7                     7

$$\frac{2 \sqrt{7} \left(- \pi + \operatorname{atan}{\left(\sqrt{7} \right)}\right)}{7} + \frac{2 \sqrt{7} \pi}{7}$$

       ___       ___ /          /  ___\\
2*pi*\/ 7    2*\/ 7 *\-pi + atan\\/ 7 //
---------- + ---------------------------
    7                     7

$$\frac{2 \sqrt{7} \left(- \pi + \operatorname{atan}{\left(\sqrt{7} \right)}\right)}{7} + \frac{2 \sqrt{7} \pi}{7}$$

2*pi*sqrt(7)/7 + 2*sqrt(7)*(-pi + atan(sqrt(7)))/7

Numerical answer [src]

0.914242542623208

0.914242542623208

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

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Identical expressions

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Integral of 1/(4-3cos(x)) dx

Limits of integration:

The graph:

Piecewise:

The solution