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Integral of sinx/(2cosx-3) dx

The solution

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

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

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

Detail solution

Let $u = 2 \cos{\left(x \right)} - 3$ .

Then let $du = - 2 \sin{\left(x \right)} dx$ 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:

$- \frac{\log{\left(2 \cos{\left(x \right)} - 3 \right)}}{2}$
Now simplify:

$- \frac{\log{\left(2 \cos{\left(x \right)} - 3 \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

  log(2)   log(3/2 - cos(1))
- ------ - -----------------
    2              2

$$- \frac{\log{\left(2 \right)}}{2} - \frac{\log{\left(\frac{3}{2} - \cos{\left(1 \right)} \right)}}{2}$$

  log(2)   log(3/2 - cos(1))
- ------ - -----------------
    2              2

$$- \frac{\log{\left(2 \right)}}{2} - \frac{\log{\left(\frac{3}{2} - \cos{\left(1 \right)} \right)}}{2}$$

-log(2)/2 - log(3/2 - cos(1))/2

Numerical answer [src]

-0.326005117250841

-0.326005117250841

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

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Integral of sinx/(2cosx-3) dx

Limits of integration:

The graph:

Piecewise:

The solution