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Integral of cos(lnx)/x^2 dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

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
  1. Let .

    Then let and substitute :

    1. There are multiple ways to do this integral.

      Method #1

      1. Let .

        Then let and substitute :

        1. The integral of a constant times a function is the constant times the integral of the function:

          1. Use integration by parts, noting that the integrand eventually repeats itself.

            1. For the integrand :

              Let and let .

              Then .

            2. For the integrand :

              Let and let .

              Then .

            3. Notice that the integrand has repeated itself, so move it to one side:

              Therefore,

          So, the result is:

        Now substitute back in:

      Method #2

      1. Use integration by parts, noting that the integrand eventually repeats itself.

        1. For the integrand :

          Let and let .

          Then .

        2. For the integrand :

          Let and let .

          Then .

        3. Notice that the integrand has repeated itself, so move it to one side:

          Therefore,

    Now substitute back in:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                              
 |                                               
 | cos(log(x))          sin(log(x))   cos(log(x))
 | ----------- dx = C + ----------- - -----------
 |       2                  2*x           2*x    
 |      x                                        
 |                                               
/                                                
$$\int \frac{\cos{\left(\log{\left(x \right)} \right)}}{x^{2}}\, dx = C + \frac{\sin{\left(\log{\left(x \right)} \right)}}{2 x} - \frac{\cos{\left(\log{\left(x \right)} \right)}}{2 x}$$
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.