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Integral of 2*cos(x)/(3-4sin(x)) dx

The solution

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

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

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

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = 3 - 4 \sin{\left(x \right)}$ .
  
  Then let $du = - 4 \cos{\left(x \right)} dx$ and substitute $- \frac{du}{2}$ :
  
  $\int \left(- \frac{1}{2 u}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{u}\, du = - \frac{\int \frac{1}{u}\, du}{2}$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    So, the result is: $- \frac{\log{\left(u \right)}}{2}$
  Now substitute $u$ back in:
  
  $- \frac{\log{\left(3 - 4 \sin{\left(x \right)} \right)}}{2}$
Method #2
1. Rewrite the integrand:
  
  $\frac{2 \cos{\left(x \right)}}{3 - 4 \sin{\left(x \right)}} = - \frac{2 \cos{\left(x \right)}}{4 \sin{\left(x \right)} - 3}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{2 \cos{\left(x \right)}}{4 \sin{\left(x \right)} - 3}\right)\, dx = - 2 \int \frac{\cos{\left(x \right)}}{4 \sin{\left(x \right)} - 3}\, dx$
  1. Let $u = 4 \sin{\left(x \right)} - 3$ .
    
    Then let $du = 4 \cos{\left(x \right)} dx$ and substitute $\frac{du}{4}$ :
    
    $\int \frac{1}{4 u}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{4}$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $\frac{\log{\left(u \right)}}{4}$
    Now substitute $u$ back in:
    
    $\frac{\log{\left(4 \sin{\left(x \right)} - 3 \right)}}{4}$
  So, the result is: $- \frac{\log{\left(4 \sin{\left(x \right)} - 3 \right)}}{2}$
Add the constant of integration:

$- \frac{\log{\left(3 - 4 \sin{\left(x \right)} \right)}}{2}+ \mathrm{constant}$

The answer is:

$- \frac{\log{\left(3 - 4 \sin{\left(x \right)} \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                       
 |                                        
 |   2*cos(x)            log(3 - 4*sin(x))
 | ------------ dx = C - -----------------
 | 3 - 4*sin(x)                  2        
 |                                        
/

$$\int \frac{2 \cos{\left(x \right)}}{3 - 4 \sin{\left(x \right)}}\, dx = C - \frac{\log{\left(3 - 4 \sin{\left(x \right)} \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

nan

$$\text{NaN}$$

nan

$$\text{NaN}$$

nan

Numerical answer [src]

1.14675182910181

1.14675182910181

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

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Identical expressions

Similar expressions

Integral of 2*cos(x)/(3-4sin(x)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2