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Identical expressions
cos(x)/(five +10sin(x))
co sinus of e of (x) divide by (5 plus 10 sinus of (x))
co sinus of e of (x) divide by (five plus 10 sinus of (x))
cosx/5+10sinx
cos(x) divide by (5+10sin(x))
cos(x)/(5+10sin(x))dx
Similar expressions
cos(x)/(5-10sin(x))
cosx/(5+10sinx)

Integral of cos(x)/(5+10sin(x)) dx

The solution

You have entered [src]

  1                 
  /                 
 |                  
 |      cos(x)      
 |  ------------- dx
 |  5 + 10*sin(x)   
 |                  
/                   
0

$$\int\limits_{0}^{1} \frac{\cos{\left(x \right)}}{10 \sin{\left(x \right)} + 5}\, dx$$

Integral(cos(x)/(5 + 10*sin(x)), (x, 0, 1))

Detail solution

Let $u = 10 \sin{\left(x \right)} + 5$ .

Then let $du = 10 \cos{\left(x \right)} dx$ and substitute $\frac{du}{10}$ :

$\int \frac{1}{10 u}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{10}$
  1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
  So, the result is: $\frac{\log{\left(u \right)}}{10}$
Now substitute $u$ back in:

$\frac{\log{\left(10 \sin{\left(x \right)} + 5 \right)}}{10}$
Add the constant of integration:

$\frac{\log{\left(10 \sin{\left(x \right)} + 5 \right)}}{10}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(10 \sin{\left(x \right)} + 5 \right)}}{10}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                         
 |                                          
 |     cos(x)             log(5 + 10*sin(x))
 | ------------- dx = C + ------------------
 | 5 + 10*sin(x)                  10        
 |                                          
/

$$\int \frac{\cos{\left(x \right)}}{10 \sin{\left(x \right)} + 5}\, dx = C + \frac{\log{\left(10 \sin{\left(x \right)} + 5 \right)}}{10}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

log(2)   log(1/2 + sin(1))
------ + -----------------
  10             10

$$\frac{\log{\left(\frac{1}{2} + \sin{\left(1 \right)} \right)}}{10} + \frac{\log{\left(2 \right)}}{10}$$

log(2)   log(1/2 + sin(1))
------ + -----------------
  10             10

$$\frac{\log{\left(\frac{1}{2} + \sin{\left(1 \right)} \right)}}{10} + \frac{\log{\left(2 \right)}}{10}$$

log(2)/10 + log(1/2 + sin(1))/10

Numerical answer [src]

0.0986913942292618

0.0986913942292618

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

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Identical expressions

Similar expressions

Integral of cos(x)/(5+10sin(x)) dx

Limits of integration:

The graph:

Piecewise:

The solution