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Identical expressions
(x+cosx)/(x+2sinx)
(x plus co sinus of e of x) divide by (x plus 2 sinus of x)
x+cosx/x+2sinx
(x+cosx) divide by (x+2sinx)
(x+cosx)/(x+2sinx)dx
Similar expressions
(x+cosx)/(x-2sinx)
(x-cosx)/(x+2sinx)

Integral of (x+cosx)/(x+2sinx) dx

The solution

You have entered [src]

 2*pi               
   /                
  |                 
  |   x + cos(x)    
  |  ------------ dx
  |  x + 2*sin(x)   
  |                 
 /                  
 pi

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

Integral((x + cos(x))/(x + 2*sin(x)), (x, pi, 2*pi))

Detail solution

Rewrite the integrand:

$\frac{x + \cos{\left(x \right)}}{x + 2 \sin{\left(x \right)}} = \frac{x}{x + 2 \sin{\left(x \right)}} + \frac{\cos{\left(x \right)}}{x + 2 \sin{\left(x \right)}}$
Integrate term-by-term:
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\int \frac{x}{x + 2 \sin{\left(x \right)}}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\int \frac{\cos{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\, dx$
The result is: $\int \frac{x}{x + 2 \sin{\left(x \right)}}\, dx + \int \frac{\cos{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\, dx$
Add the constant of integration:

$\int \frac{x}{x + 2 \sin{\left(x \right)}}\, dx + \int \frac{\cos{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\, dx+ \mathrm{constant}$

The answer is:

$\int \frac{x}{x + 2 \sin{\left(x \right)}}\, dx + \int \frac{\cos{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\, dx+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                        /                    /               
 |                        |                    |                
 |  x + cos(x)            |      x             |    cos(x)      
 | ------------ dx = C +  | ------------ dx +  | ------------ dx
 | x + 2*sin(x)           | x + 2*sin(x)       | x + 2*sin(x)   
 |                        |                    |                
/                        /                    /

$$\int \frac{x + \cos{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\, dx = C + \int \frac{x}{x + 2 \sin{\left(x \right)}}\, dx + \int \frac{\cos{\left(x \right)}}{x + 2 \sin{\left(x \right)}}\, dx$$

Simplify

The answer [src]

 2*pi               
   /                
  |                 
  |   x + cos(x)    
  |  ------------ dx
  |  x + 2*sin(x)   
  |                 
 /                  
 pi

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

 2*pi               
   /                
  |                 
  |   x + cos(x)    
  |  ------------ dx
  |  x + 2*sin(x)   
  |                 
 /                  
 pi

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

Integral((x + cos(x))/(x + 2*sin(x)), (x, pi, 2*pi))

Numerical answer [src]

4.33494741276426

4.33494741276426

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

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Identical expressions

Similar expressions

Integral of (x+cosx)/(x+2sinx) dx

Limits of integration:

The graph:

Piecewise:

The solution