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cos(lnx)/x^2dx

Integral of cos(lnx)/x^2 dx

The solution

You have entered [src]

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

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

Integral(cos(log(x))/x^2, (x, 0, 1))

Detail solution

Let $u = \log{\left(x \right)}$ .

Then let $du = \frac{dx}{x}$ and substitute $du$ :

$\int e^{- u} \cos{\left(u \right)}\, du$
1. There are multiple ways to do this integral.
  Method #1
  1. Let $u = - u$ .
    
    Then let $du = - du$ and substitute $- du$ :
    
    $\int \left(- e^{u} \cos{\left(u \right)}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int e^{u} \cos{\left(u \right)}\, du = - \int e^{u} \cos{\left(u \right)}\, du$
      
      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:
    
    $\frac{e^{- u} \sin{\left(u \right)}}{2} - \frac{e^{- u} \cos{\left(u \right)}}{2}$
  Method #2
  1. Use integration by parts, noting that the integrand eventually repeats itself.
    1. 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 $\int e^{- u} \cos{\left(u \right)}\, du = - \int e^{- u} \sin{\left(u \right)}\, du - e^{- u} \cos{\left(u \right)}$ .
    2. 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 $\int e^{- u} \cos{\left(u \right)}\, du = \int \left(- e^{- u} \cos{\left(u \right)}\right)\, du + e^{- u} \sin{\left(u \right)} - e^{- u} \cos{\left(u \right)}$ .
    3. 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,
      
      $\int e^{- u} \cos{\left(u \right)}\, du = \frac{e^{- u} \sin{\left(u \right)}}{2} - \frac{e^{- u} \cos{\left(u \right)}}{2}$
Now substitute $u$ back in:

$\frac{\sin{\left(\log{\left(x \right)} \right)}}{2 x} - \frac{\cos{\left(\log{\left(x \right)} \right)}}{2 x}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

<-oo, oo>

$$\left\langle -\infty, \infty\right\rangle$$

<-oo, oo>

$$\left\langle -\infty, \infty\right\rangle$$

AccumBounds(-oo, oo)

Numerical answer [src]

7.49656888054513e+18

7.49656888054513e+18

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

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Identical expressions

Integral of cos(lnx)/x^2 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2