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Identical expressions
(2cos(ln(x)+ five))/x
(2 co sinus of e of (ln(x) plus 5)) divide by x
(2 co sinus of e of (ln(x) plus five)) divide by x
2coslnx+5/x
(2cos(ln(x)+5)) divide by x
(2cos(ln(x)+5))/xdx
Similar expressions
(2cos(ln(x)-5))/x

Integral of (2cos(ln(x)+5))/x dx

The solution

You have entered [src]

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

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

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

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \log{\left(x \right)} + 5$ .
  
  Then let $du = \frac{dx}{x}$ and substitute $2 du$ :
  
  $\int 2 \cos{\left(u \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = 2 \int \cos{\left(u \right)}\, du$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $2 \sin{\left(u \right)}$
  Now substitute $u$ back in:
  
  $2 \sin{\left(\log{\left(x \right)} + 5 \right)}$
Method #2
1. Let $u = \frac{1}{x}$ .
  
  Then let $du = - \frac{dx}{x^{2}}$ and substitute $- 2 du$ :
  
  $\int \left(- \frac{2 \cos{\left(\log{\left(\frac{1}{u} \right)} + 5 \right)}}{u}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos{\left(\log{\left(\frac{1}{u} \right)} + 5 \right)}}{u}\, du = - 2 \int \frac{\cos{\left(\log{\left(\frac{1}{u} \right)} + 5 \right)}}{u}\, du$
    1. Let $u = \frac{1}{u}$ .
      
      Then let $du = - \frac{du}{u^{2}}$ and substitute $- du$ :
      
      $\int \left(- \frac{\cos{\left(\log{\left(u \right)} + 5 \right)}}{u}\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(\log{\left(u \right)} + 5 \right)}}{u}\, du = - \int \frac{\cos{\left(\log{\left(u \right)} + 5 \right)}}{u}\, du$
      
      Let $u = \log{\left(u \right)} + 5$ .
      
      Then let $du = \frac{du}{u}$ and substitute $du$ :
      
      $\int \cos{\left(u \right)}\, du$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $\sin{\left(\log{\left(u \right)} + 5 \right)}$
      
      So, the result is: $- \sin{\left(\log{\left(u \right)} + 5 \right)}$
      
      Now substitute $u$ back in:
      
      $\sin{\left(\log{\left(u \right)} - 5 \right)}$
    So, the result is: $- 2 \sin{\left(\log{\left(u \right)} - 5 \right)}$
  Now substitute $u$ back in:
  
  $2 \sin{\left(\log{\left(x \right)} + 5 \right)}$
Now simplify:

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

$2 \sin{\left(\log{\left(x \right)} + 5 \right)}+ \mathrm{constant}$

The answer is:

$2 \sin{\left(\log{\left(x \right)} + 5 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

<-2 + 2*sin(5), 2 + 2*sin(5)>

$$\left\langle -2 + 2 \sin{\left(5 \right)}, 2 \sin{\left(5 \right)} + 2\right\rangle$$

<-2 + 2*sin(5), 2 + 2*sin(5)>

$$\left\langle -2 + 2 \sin{\left(5 \right)}, 2 \sin{\left(5 \right)} + 2\right\rangle$$

AccumBounds(-2 + 2*sin(5), 2 + 2*sin(5))

Numerical answer [src]

0.0895756425026216

0.0895756425026216

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

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Identical expressions

Similar expressions

Integral of (2cos(ln(x)+5))/x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2