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Identical expressions
cos(lnx)*(d*x)/x
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cos(lnx)*(d*x)/xdx

Integral of cos(lnx)(dx)/x dx

The solution

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

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

Integral((cos(log(x))*(d*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 $d du$ :
  
  $\int d e^{u} \cos{\left(u \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int e^{u} \cos{\left(u \right)}\, du = d \int e^{u} \cos{\left(u \right)}\, du$
    1. Use integration by parts, noting that the integrand eventually repeats itself.
      
      For the integrand $e^{u} \cos{\left(u \right)}$ :
      
      Let $u{\left(u \right)} = \cos{\left(u \right)}$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
      
      Then $\int e^{u} \cos{\left(u \right)}\, du = e^{u} \cos{\left(u \right)} - \int \left(- e^{u} \sin{\left(u \right)}\right)\, du$ .
      
      For the integrand $- e^{u} \sin{\left(u \right)}$ :
      
      Let $u{\left(u \right)} = - \sin{\left(u \right)}$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
      
      Then $\int e^{u} \cos{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} + e^{u} \cos{\left(u \right)} + \int \left(- e^{u} \cos{\left(u \right)}\right)\, du$ .
      
      Notice that the integrand has repeated itself, so move it to one side:
      
      $2 \int e^{u} \cos{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} + e^{u} \cos{\left(u \right)}$
      
      Therefore,
      
      $\int e^{u} \cos{\left(u \right)}\, du = \frac{e^{u} \sin{\left(u \right)}}{2} + \frac{e^{u} \cos{\left(u \right)}}{2}$
    So, the result is: $d \left(\frac{e^{u} \sin{\left(u \right)}}{2} + \frac{e^{u} \cos{\left(u \right)}}{2}\right)$
  Now substitute $u$ back in:
  
  $d \left(\frac{x \sin{\left(\log{\left(x \right)} \right)}}{2} + \frac{x \cos{\left(\log{\left(x \right)} \right)}}{2}\right)$
Method #2
1. Let $u = \frac{1}{x}$ .
  
  Then let $du = - \frac{dx}{x^{2}}$ and substitute $- d du$ :
  
  $\int \left(- \frac{d \cos{\left(\log{\left(\frac{1}{u} \right)} \right)}}{u^{2}}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos{\left(\log{\left(\frac{1}{u} \right)} \right)}}{u^{2}}\, du = - d \int \frac{\cos{\left(\log{\left(\frac{1}{u} \right)} \right)}}{u^{2}}\, du$
    1. Let $u = \log{\left(\frac{1}{u} \right)}$ .
      
      Then let $du = - \frac{du}{u}$ and substitute $- du$ :
      
      $\int \left(- e^{u} \cos{\left(u \right)}\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int e^{u} \cos{\left(u \right)}\, du = - \int e^{u} \cos{\left(u \right)}\, du$
      
      Use integration by parts, noting that the integrand eventually repeats itself.
      
      For the integrand $e^{u} \cos{\left(u \right)}$ :
      
      Let $u{\left(u \right)} = \cos{\left(u \right)}$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
      
      Then $\int e^{u} \cos{\left(u \right)}\, du = e^{u} \cos{\left(u \right)} - \int \left(- e^{u} \sin{\left(u \right)}\right)\, du$ .
      
      For the integrand $- e^{u} \sin{\left(u \right)}$ :
      
      Let $u{\left(u \right)} = - \sin{\left(u \right)}$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
      
      Then $\int e^{u} \cos{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} + e^{u} \cos{\left(u \right)} + \int \left(- e^{u} \cos{\left(u \right)}\right)\, du$ .
      
      Notice that the integrand has repeated itself, so move it to one side:
      
      $2 \int e^{u} \cos{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} + e^{u} \cos{\left(u \right)}$
      
      Therefore,
      
      $\int e^{u} \cos{\left(u \right)}\, du = \frac{e^{u} \sin{\left(u \right)}}{2} + \frac{e^{u} \cos{\left(u \right)}}{2}$
      
      So, the result is: $- \frac{e^{u} \sin{\left(u \right)}}{2} - \frac{e^{u} \cos{\left(u \right)}}{2}$
      
      Now substitute $u$ back in:
      
      $- \frac{\sin{\left(\log{\left(\frac{1}{u} \right)} \right)}}{2 u} - \frac{\cos{\left(\log{\left(\frac{1}{u} \right)} \right)}}{2 u}$
    So, the result is: $- d \left(- \frac{\sin{\left(\log{\left(\frac{1}{u} \right)} \right)}}{2 u} - \frac{\cos{\left(\log{\left(\frac{1}{u} \right)} \right)}}{2 u}\right)$
  Now substitute $u$ back in:
  
  $- d \left(- \frac{x \sin{\left(\log{\left(x \right)} \right)}}{2} - \frac{x \cos{\left(\log{\left(x \right)} \right)}}{2}\right)$
Now simplify:

$\frac{\sqrt{2} d x \sin{\left(\log{\left(x \right)} + \frac{\pi}{4} \right)}}{2}$
Add the constant of integration:

$\frac{\sqrt{2} d x \sin{\left(\log{\left(x \right)} + \frac{\pi}{4} \right)}}{2}+ \mathrm{constant}$

The answer is:

$\frac{\sqrt{2} d x \sin{\left(\log{\left(x \right)} + \frac{\pi}{4} \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

d
-
2

$$\frac{d}{2}$$

d
-
2

$$\frac{d}{2}$$

d/2

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

Other calculators

How to use it?

Identical expressions

Integral of cos(lnx)(dx)/x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Other calculators

How to use it?

Identical expressions

Integral of cos(lnx)*(d*x)/x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Integral of cos(lnx)(dx)/x dx