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Identical expressions
(e*cos(ln(x)))/x
(e multiply by co sinus of e of (ln(x))) divide by x
(ecos(ln(x)))/x
ecoslnx/x
(e*cos(ln(x))) divide by x
(e*cos(ln(x)))/xdx

Integral of (e*cos(ln(x)))/x dx

The solution

You have entered [src]

  1                 
  /                 
 |                  
 |  E*cos(log(x))   
 |  ------------- dx
 |        x         
 |                  
/                   
0

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

Integral((E*cos(log(x)))/x, (x, 0, 1))

Detail solution

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

-<-1, 1>*E

$$- \left\langle -1, 1\right\rangle e$$

-<-1, 1>*E

$$- \left\langle -1, 1\right\rangle e$$

-AccumBounds(-1, 1)*E

Numerical answer [src]

0.299165685009485

0.299165685009485

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

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How to use it?

Identical expressions

Integral of (e*cos(ln(x)))/x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2