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Similar expressions
sinx/(3-2cosx)½

Integral of sinx/(3+2cosx)½ dx

The solution

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

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

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

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{\frac{1}{2 \cos{\left(x \right)} + 3} \sin{\left(x \right)}}{2}\, dx = \frac{\int \frac{\sin{\left(x \right)}}{2 \cos{\left(x \right)} + 3}\, dx}{2}$
1. 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:
  
  $- \frac{\log{\left(2 \cos{\left(x \right)} + 3 \right)}}{2}$
So, the result is: $- \frac{\log{\left(2 \cos{\left(x \right)} + 3 \right)}}{4}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

$$\int \frac{\frac{1}{2 \cos{\left(x \right)} + 3} \sin{\left(x \right)}}{2}\, dx = C - \frac{\log{\left(2 \cos{\left(x \right)} + 3 \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  log(3/2 + cos(1))   log(5/2)
- ----------------- + --------
          4              4

$$- \frac{\log{\left(\cos{\left(1 \right)} + \frac{3}{2} \right)}}{4} + \frac{\log{\left(\frac{5}{2} \right)}}{4}$$

  log(3/2 + cos(1))   log(5/2)
- ----------------- + --------
          4              4

$$- \frac{\log{\left(\cos{\left(1 \right)} + \frac{3}{2} \right)}}{4} + \frac{\log{\left(\frac{5}{2} \right)}}{4}$$

-log(3/2 + cos(1))/4 + log(5/2)/4

Numerical answer [src]

0.0507981864614772

0.0507981864614772

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

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Identical expressions

Similar expressions

Integral of sinx/(3+2cosx)½ dx

Limits of integration:

The graph:

Piecewise:

The solution