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

1. 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 $e du$:

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

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

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

         1. The integral of cosine is sine:

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

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

      Now substitute $u$ back in:

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

   Method #2

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

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

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

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

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

         1. Let $u = \frac{1}{u}$.

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

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

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

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

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

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

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

                  1. The integral of cosine is sine:

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

                  Now substitute $u$ back in:

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

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

            Now substitute $u$ back in:

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

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

      Now substitute $u$ back in:

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

2. Add the constant of integration:

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

The answer is:

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