Integral of 2*ln(x)-5*x^5 dx

1. Integrate term-by-term:

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

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

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

         $$\int x^{5}\, dx = \frac{x^{6}}{6}$$

      So, the result is: $- \frac{5 x^{6}}{6}$

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

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

      1. Use integration by parts:

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

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

         Then $\operatorname{du}{\left(x \right)} = \frac{1}{x}$.

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

         1. The integral of a constant is the constant times the variable of integration:

            $$\int 1\, dx = x$$

         Now evaluate the sub-integral.

      2. The integral of a constant is the constant times the variable of integration:

         $$\int 1\, dx = x$$

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

   The result is: $- \frac{5 x^{6}}{6} + 2 x \log{\left(x \right)} - 2 x$

2. Now simplify:

   $$\frac{x \left(- 5 x^{5} + 12 \log{\left(x \right)} - 12\right)}{6}$$

3. Add the constant of integration:

   $$\frac{x \left(- 5 x^{5} + 12 \log{\left(x \right)} - 12\right)}{6}+ \mathrm{constant}$$

The answer is:

$$\frac{x \left(- 5 x^{5} + 12 \log{\left(x \right)} - 12\right)}{6}+ \mathrm{constant}$$