Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of x^2*e^3x
Integral of cos2xsinx
Integral of (-1)^x
Integral of (x^3-1)/(x+3)
Identical expressions
(two +Cosx)/(two -Sinx)
(2 plus Cosx) divide by (2 minus Sinx)
(two plus Cosx) divide by (two minus Sinx)
2+Cosx/2-Sinx
(2+Cosx) divide by (2-Sinx)
(2+Cosx)/(2-Sinx)dx
Similar expressions
(2-Cosx)/(2-Sinx)
(2+Cosx)/(2+Sinx)

Integral of (2+Cosx)/(2-Sinx) dx

The solution

You have entered [src]

  1              
  /              
 |               
 |  2 + cos(x)   
 |  ---------- dx
 |  2 - sin(x)   
 |               
/                
0

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

Integral((2 + cos(x))/(2 - sin(x)), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{\cos{\left(x \right)} + 2}{2 - \sin{\left(x \right)}} = - \frac{\cos{\left(x \right)} + 2}{\sin{\left(x \right)} - 2}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\cos{\left(x \right)} + 2}{\sin{\left(x \right)} - 2}\right)\, dx = - \int \frac{\cos{\left(x \right)} + 2}{\sin{\left(x \right)} - 2}\, dx$
  1. Rewrite the integrand:
    
    $\frac{\cos{\left(x \right)} + 2}{\sin{\left(x \right)} - 2} = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)} - 2} + \frac{2}{\sin{\left(x \right)} - 2}$
  2. Integrate term-by-term:
    1. Let $u = \sin{\left(x \right)} - 2$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(\sin{\left(x \right)} - 2 \right)}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{2}{\sin{\left(x \right)} - 2}\, dx = 2 \int \frac{1}{\sin{\left(x \right)} - 2}\, dx$
      
      Don't know the steps in finding this integral.
      
      But the integral is
      
      $- \frac{2 \sqrt{3} \left(\operatorname{atan}{\left(\frac{2 \sqrt{3} \tan{\left(\frac{x}{2} \right)}}{3} - \frac{\sqrt{3}}{3} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{3}$
      
      So, the result is: $- \frac{4 \sqrt{3} \left(\operatorname{atan}{\left(\frac{2 \sqrt{3} \tan{\left(\frac{x}{2} \right)}}{3} - \frac{\sqrt{3}}{3} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{3}$
    The result is: $- \frac{4 \sqrt{3} \left(\operatorname{atan}{\left(\frac{2 \sqrt{3} \tan{\left(\frac{x}{2} \right)}}{3} - \frac{\sqrt{3}}{3} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{3} + \log{\left(\sin{\left(x \right)} - 2 \right)}$
  So, the result is: $\frac{4 \sqrt{3} \left(\operatorname{atan}{\left(\frac{2 \sqrt{3} \tan{\left(\frac{x}{2} \right)}}{3} - \frac{\sqrt{3}}{3} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{3} - \log{\left(\sin{\left(x \right)} - 2 \right)}$
Method #2
1. Rewrite the integrand:
  
  $\frac{\cos{\left(x \right)} + 2}{2 - \sin{\left(x \right)}} = \frac{\cos{\left(x \right)}}{2 - \sin{\left(x \right)}} + \frac{2}{2 - \sin{\left(x \right)}}$
2. Integrate term-by-term:
  1. Let $u = 2 - \sin{\left(x \right)}$ .
    
    Then let $du = - \cos{\left(x \right)} dx$ and substitute $- du$ :
    
    $\int \left(- \frac{1}{u}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{u}\, du = - \int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $- \log{\left(u \right)}$
    Now substitute $u$ back in:
    
    $- \log{\left(2 - \sin{\left(x \right)} \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{2}{2 - \sin{\left(x \right)}}\, dx = 2 \int \frac{1}{2 - \sin{\left(x \right)}}\, dx$
    1. Rewrite the integrand:
      
      $\frac{1}{2 - \sin{\left(x \right)}} = - \frac{1}{\sin{\left(x \right)} - 2}$
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{\sin{\left(x \right)} - 2}\right)\, dx = - \int \frac{1}{\sin{\left(x \right)} - 2}\, dx$
      
      Don't know the steps in finding this integral.
      
      But the integral is
      
      $- \frac{2 \sqrt{3} \left(\operatorname{atan}{\left(\frac{2 \sqrt{3} \tan{\left(\frac{x}{2} \right)}}{3} - \frac{\sqrt{3}}{3} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{3}$
      
      So, the result is: $\frac{2 \sqrt{3} \left(\operatorname{atan}{\left(\frac{2 \sqrt{3} \tan{\left(\frac{x}{2} \right)}}{3} - \frac{\sqrt{3}}{3} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{3}$
    So, the result is: $\frac{4 \sqrt{3} \left(\operatorname{atan}{\left(\frac{2 \sqrt{3} \tan{\left(\frac{x}{2} \right)}}{3} - \frac{\sqrt{3}}{3} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{3}$
  The result is: $\frac{4 \sqrt{3} \left(\operatorname{atan}{\left(\frac{2 \sqrt{3} \tan{\left(\frac{x}{2} \right)}}{3} - \frac{\sqrt{3}}{3} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{3} - \log{\left(2 - \sin{\left(x \right)} \right)}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

                                              /          /  ___       ___         \\                                            
                                          ___ |          |\/ 3    2*\/ 3 *tan(1/2)||                                            
                                      4*\/ 3 *|-pi - atan|----- - ----------------||           ___                              
     /                      2     \           \          \  3            3        //   14*pi*\/ 3                /       2     \
- log\4 - 4*tan(1/2) + 4*tan (1/2)/ + ---------------------------------------------- + ----------- + log(4) + log\1 + tan (1/2)/
                                                            3                               9

$$\frac{4 \sqrt{3} \left(- \pi - \operatorname{atan}{\left(- \frac{2 \sqrt{3} \tan{\left(\frac{1}{2} \right)}}{3} + \frac{\sqrt{3}}{3} \right)}\right)}{3} - \log{\left(- 4 \tan{\left(\frac{1}{2} \right)} + 4 \tan^{2}{\left(\frac{1}{2} \right)} + 4 \right)} + \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} + \log{\left(4 \right)} + \frac{14 \sqrt{3} \pi}{9}$$

                                              /          /  ___       ___         \\                                            
                                          ___ |          |\/ 3    2*\/ 3 *tan(1/2)||                                            
                                      4*\/ 3 *|-pi - atan|----- - ----------------||           ___                              
     /                      2     \           \          \  3            3        //   14*pi*\/ 3                /       2     \
- log\4 - 4*tan(1/2) + 4*tan (1/2)/ + ---------------------------------------------- + ----------- + log(4) + log\1 + tan (1/2)/
                                                            3                               9

-log(4 - 4*tan(1/2) + 4*tan(1/2)^2) + 4*sqrt(3)*(-pi - atan(sqrt(3)/3 - 2*sqrt(3)*tan(1/2)/3))/3 + 14*pi*sqrt(3)/9 + log(4) + log(1 + tan(1/2)^2)

Numerical answer [src]

1.87855150230211

1.87855150230211

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (2+Cosx)/(2-Sinx) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2