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Identical expressions
(sinx)/(five -3cosx)
( sinus of x) divide by (5 minus 3 co sinus of e of x)
( sinus of x) divide by (five minus 3 co sinus of e of x)
sinx/5-3cosx
(sinx) divide by (5-3cosx)
(sinx)/(5-3cosx)dx
Similar expressions
(sin(x))/(5-3*cos(x))
(sinx)/(5+3cosx)

Solving integrals/ (sinx)/(5-3cosx)

Integral of (sinx)/(5-3cosx) dx

The solution

You have entered [src]

 pi                
 --                
 3                 
  /                
 |                 
 |     sin(x)      
 |  ------------ dx
 |  5 - 3*cos(x)   
 |                 
/                  
pi

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

Integral(sin(x)/(5 - 3*cos(x)), (x, pi, pi/3))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = 5 - 3 \cos{\left(x \right)}$ .
  
  Then let $du = 3 \sin{\left(x \right)} dx$ and substitute $\frac{du}{3}$ :
  
  $\int \frac{1}{3 u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{3}$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    So, the result is: $\frac{\log{\left(u \right)}}{3}$
  Now substitute $u$ back in:
  
  $\frac{\log{\left(5 - 3 \cos{\left(x \right)} \right)}}{3}$
Method #2
1. Rewrite the integrand:
  
  $\frac{\sin{\left(x \right)}}{5 - 3 \cos{\left(x \right)}} = - \frac{\sin{\left(x \right)}}{3 \cos{\left(x \right)} - 5}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\sin{\left(x \right)}}{3 \cos{\left(x \right)} - 5}\right)\, dx = - \int \frac{\sin{\left(x \right)}}{3 \cos{\left(x \right)} - 5}\, dx$
  1. Let $u = 3 \cos{\left(x \right)} - 5$ .
    
    Then let $du = - 3 \sin{\left(x \right)} dx$ and substitute $- \frac{du}{3}$ :
    
    $\int \left(- \frac{1}{3 u}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{u}\, du = - \frac{\int \frac{1}{u}\, du}{3}$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $- \frac{\log{\left(u \right)}}{3}$
    Now substitute $u$ back in:
    
    $- \frac{\log{\left(3 \cos{\left(x \right)} - 5 \right)}}{3}$
  So, the result is: $\frac{\log{\left(3 \cos{\left(x \right)} - 5 \right)}}{3}$
Add the constant of integration:

$\frac{\log{\left(5 - 3 \cos{\left(x \right)} \right)}}{3}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(5 - 3 \cos{\left(x \right)} \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                       
 |                                        
 |    sin(x)             log(5 - 3*cos(x))
 | ------------ dx = C + -----------------
 | 5 - 3*cos(x)                  3        
 |                                        
/

$$\int \frac{\sin{\left(x \right)}}{5 - 3 \cos{\left(x \right)}}\, dx = C + \frac{\log{\left(5 - 3 \cos{\left(x \right)} \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  log(8/3)   log(7/6)
- -------- + --------
     3          3

$$- \frac{\log{\left(\frac{8}{3} \right)}}{3} + \frac{\log{\left(\frac{7}{6} \right)}}{3}$$

  log(8/3)   log(7/6)
- -------- + --------
     3          3

$$- \frac{\log{\left(\frac{8}{3} \right)}}{3} + \frac{\log{\left(\frac{7}{6} \right)}}{3}$$

-log(8/3)/3 + log(7/6)/3

Numerical answer [src]

-0.275559524394823

-0.275559524394823

The graph

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (sinx)/(5-3cosx) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2