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

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

The solution

You have entered [src]

 oo              
  /              
 |               
 |    cos(x)     
 |  ---------- dx
 |  5 + sin(x)   
 |               
/                
0

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

Integral(cos(x)/(5 + sin(x)), (x, 0, oo))

Detail solution

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

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

$\int \frac{1}{u}\, du$
1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
Now substitute $u$ back in:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

<-log(5) + log(4), -log(5) + log(6)>

$$\left\langle - \log{\left(5 \right)} + \log{\left(4 \right)}, - \log{\left(5 \right)} + \log{\left(6 \right)}\right\rangle$$

<-log(5) + log(4), -log(5) + log(6)>

$$\left\langle - \log{\left(5 \right)} + \log{\left(4 \right)}, - \log{\left(5 \right)} + \log{\left(6 \right)}\right\rangle$$

AccumBounds(-log(5) + log(4), -log(5) + log(6))

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution