Mister Exam

Other calculators

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                 
  /                 
 |                  
 |  E*cos(log(x))   
 |  ------------- dx
 |        x         
 |                  
/                   
0                   
01ecos(log(x))xdx\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
  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 edue du:

      ecos(u)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:

        cos(u)du=ecos(u)du\int \cos{\left(u \right)}\, du = e \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: esin(u)e \sin{\left(u \right)}

      Now substitute uu back in:

      esin(log(x))e \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 edu- e du:

      (ecos(log(1u))u)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:

        cos(log(1u))udu=ecos(log(1u))udu\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=1uu = \frac{1}{u}.

          Then let du=duu2du = - \frac{du}{u^{2}} and substitute du- du:

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

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

              Then let du=duudu = \frac{du}{u} 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(u))\sin{\left(\log{\left(u \right)} \right)}

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

          Now substitute uu back in:

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

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

      Now substitute uu back in:

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

  2. Add the constant of integration:

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


The answer is:

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

The answer (Indefinite) [src]
  /                                    
 |                                     
 | E*cos(log(x))                       
 | ------------- dx = C + E*sin(log(x))
 |       x                             
 |                                     
/                                      
ecos(log(x))xdx=C+esin(log(x))\int \frac{e \cos{\left(\log{\left(x \right)} \right)}}{x}\, dx = C + e \sin{\left(\log{\left(x \right)} \right)}
The answer [src]
-<-1, 1>*E
1,1e- \left\langle -1, 1\right\rangle e
=
=
-<-1, 1>*E
1,1e- \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.