Integral of log(2x^5)/x^2 dx

1. There are multiple ways to do this integral.

   Method #1

   1. Rewrite the integrand:

      $$\frac{\log{\left(2 x^{5} \right)}}{x^{2}} = \frac{\log{\left(x^{5} \right)} + \log{\left(2 \right)}}{x^{2}}$$

   2. Rewrite the integrand:

      $$\frac{\log{\left(x^{5} \right)} + \log{\left(2 \right)}}{x^{2}} = \frac{\log{\left(x^{5} \right)}}{x^{2}} + \frac{\log{\left(2 \right)}}{x^{2}}$$

   3. Integrate term-by-term:

      1. Use integration by parts:

         $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

         Let $u{\left(x \right)} = \log{\left(x^{5} \right)}$ and let $\operatorname{dv}{\left(x \right)} = \frac{1}{x^{2}}$.

         Then $\operatorname{du}{\left(x \right)} = \frac{5}{x}$.

         To find $v{\left(x \right)}$:

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}$$

         Now evaluate the sub-integral.

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

         $$\int \left(- \frac{5}{x^{2}}\right)\, dx = - 5 \int \frac{1}{x^{2}}\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}$$

         So, the result is: $\frac{5}{x}$

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

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

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}$$

         So, the result is: $- \frac{\log{\left(2 \right)}}{x}$

      The result is: $- \frac{\log{\left(x^{5} \right)}}{x} - \frac{5}{x} - \frac{\log{\left(2 \right)}}{x}$

   Method #2

   1. Use integration by parts:

      $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

      Let $u{\left(x \right)} = \log{\left(2 x^{5} \right)}$ and let $\operatorname{dv}{\left(x \right)} = \frac{1}{x^{2}}$.

      Then $\operatorname{du}{\left(x \right)} = \frac{5}{x}$.

      To find $v{\left(x \right)}$:

      1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

         $$\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}$$

      Now evaluate the sub-integral.

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

      $$\int \left(- \frac{5}{x^{2}}\right)\, dx = - 5 \int \frac{1}{x^{2}}\, dx$$

      1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

         $$\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}$$

      So, the result is: $\frac{5}{x}$

   Method #3

   1. Rewrite the integrand:

      $$\frac{\log{\left(2 x^{5} \right)}}{x^{2}} = \frac{\log{\left(x^{5} \right)}}{x^{2}} + \frac{\log{\left(2 \right)}}{x^{2}}$$

   2. Integrate term-by-term:

      1. Use integration by parts:

         $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

         Let $u{\left(x \right)} = \log{\left(x^{5} \right)}$ and let $\operatorname{dv}{\left(x \right)} = \frac{1}{x^{2}}$.

         Then $\operatorname{du}{\left(x \right)} = \frac{5}{x}$.

         To find $v{\left(x \right)}$:

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}$$

         Now evaluate the sub-integral.

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

         $$\int \left(- \frac{5}{x^{2}}\right)\, dx = - 5 \int \frac{1}{x^{2}}\, dx$$

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}$$

         So, the result is: $\frac{5}{x}$

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

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

         1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}$$

         So, the result is: $- \frac{\log{\left(2 \right)}}{x}$

      The result is: $- \frac{\log{\left(x^{5} \right)}}{x} - \frac{5}{x} - \frac{\log{\left(2 \right)}}{x}$

2. Now simplify:

   $$\frac{- \log{\left(x^{5} \right)} - 5 - \log{\left(2 \right)}}{x}$$

3. Add the constant of integration:

   $$\frac{- \log{\left(x^{5} \right)} - 5 - \log{\left(2 \right)}}{x}+ \mathrm{constant}$$

The answer is:

$$\frac{- \log{\left(x^{5} \right)} - 5 - \log{\left(2 \right)}}{x}+ \mathrm{constant}$$