Mister Exam

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

Limits of integration:

from to
v

The graph:

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Piecewise:

The solution

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  1               
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 |  cos(log(x))   
 |  ----------- dx
 |       x        
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01cos(log(x))xdx\int\limits_{0}^{1} \frac{\cos{\left(\log{\left(x \right)} \right)}}{x}\, dx
Integral(cos(log(x))/x, (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

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

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

      cos(u)du\int \cos{\left(u \right)}\, du

      1. The integral of cosine is sine:

        cos(u)du=sin(u)\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}

      Now substitute uu back in:

      sin(log(x))\sin{\left(\log{\left(x \right)} \right)}

    Method #2

    1. Let u=1xu = \frac{1}{x}.

      Then let du=dxx2du = - \frac{dx}{x^{2}} and substitute du- du:

      (cos(log(1u))u)du\int \left(- \frac{\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:

        cos(log(1u))udu=cos(log(1u))udu\int \frac{\cos{\left(\log{\left(\frac{1}{u} \right)} \right)}}{u}\, du = - \int \frac{\cos{\left(\log{\left(\frac{1}{u} \right)} \right)}}{u}\, du

        1. Let u=log(1u)u = \log{\left(\frac{1}{u} \right)}.

          Then let du=duudu = - \frac{du}{u} and substitute du- du:

          (cos(u))du\int \left(- \cos{\left(u \right)}\right)\, du

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

            cos(u)du=cos(u)du\int \cos{\left(u \right)}\, du = - \int \cos{\left(u \right)}\, du

            1. The integral of cosine is sine:

              cos(u)du=sin(u)\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}

            So, the result is: sin(u)- \sin{\left(u \right)}

          Now substitute uu back in:

          sin(log(1u))- \sin{\left(\log{\left(\frac{1}{u} \right)} \right)}

        So, the result is: sin(log(1u))\sin{\left(\log{\left(\frac{1}{u} \right)} \right)}

      Now substitute uu back in:

      sin(log(x))\sin{\left(\log{\left(x \right)} \right)}

  2. Add the constant of integration:

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


The answer is:

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

The answer (Indefinite) [src]
  /                                
 |                                 
 | cos(log(x))                     
 | ----------- dx = C + sin(log(x))
 |      x                          
 |                                 
/                                  
cos(log(x))xdx=C+sin(log(x))\int \frac{\cos{\left(\log{\left(x \right)} \right)}}{x}\, dx = C + \sin{\left(\log{\left(x \right)} \right)}
The answer [src]
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1,1\left\langle -1, 1\right\rangle
=
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1,1\left\langle -1, 1\right\rangle
AccumBounds(-1, 1)
Numerical answer [src]
0.110056905018961
0.110056905018961

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