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Identical expressions
sinx/(five +2cosx)
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sinus of x divide by (five plus 2 co sinus of e of x)
sinx/5+2cosx
sinx divide by (5+2cosx)
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Similar expressions
sinx/(5-2cosx)

Solving integrals/ sinx/(5+2cosx)

Integral of sinx/(5+2cosx) dx

The solution

You have entered [src]

  1                
  /                
 |                 
 |     sin(x)      
 |  ------------ dx
 |  5 + 2*cos(x)   
 |                 
/                  
0

$$\int\limits_{0}^{1} \frac{\sin{\left(x \right)}}{2 \cos{\left(x \right)} + 5}\, dx$$

Integral(sin(x)/(5 + 2*cos(x)), (x, 0, 1))

Detail solution

Let $u = 2 \cos{\left(x \right)} + 5$ .

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:

$- \frac{\log{\left(2 \cos{\left(x \right)} + 5 \right)}}{2}$
Add the constant of integration:

$- \frac{\log{\left(2 \cos{\left(x \right)} + 5 \right)}}{2}+ \mathrm{constant}$

The answer is:

$- \frac{\log{\left(2 \cos{\left(x \right)} + 5 \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                       
 |                                        
 |    sin(x)             log(5 + 2*cos(x))
 | ------------ dx = C - -----------------
 | 5 + 2*cos(x)                  2        
 |                                        
/

$$\int \frac{\sin{\left(x \right)}}{2 \cos{\left(x \right)} + 5}\, dx = C - \frac{\log{\left(2 \cos{\left(x \right)} + 5 \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

log(7/2)   log(5/2 + cos(1))
-------- - -----------------
   2               2

$$- \frac{\log{\left(\cos{\left(1 \right)} + \frac{5}{2} \right)}}{2} + \frac{\log{\left(\frac{7}{2} \right)}}{2}$$

log(7/2)   log(5/2 + cos(1))
-------- - -----------------
   2               2

$$- \frac{\log{\left(\cos{\left(1 \right)} + \frac{5}{2} \right)}}{2} + \frac{\log{\left(\frac{7}{2} \right)}}{2}$$

log(7/2)/2 - log(5/2 + cos(1))/2

Numerical answer [src]

0.0704030076507772

0.0704030076507772

The graph

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

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Identical expressions

Similar expressions

Integral of sinx/(5+2cosx) dx

Limits of integration:

The graph:

Piecewise:

The solution