Integral of xln(3x-5) dx

1. Use integration by parts:

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

   Let $u{\left(x \right)} = \log{\left(3 x - 5 \right)}$ and let $\operatorname{dv}{\left(x \right)} = x$.

   Then $\operatorname{du}{\left(x \right)} = \frac{3}{3 x - 5}$.

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

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

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

   Now evaluate the sub-integral.

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

   $$\int \frac{3 x^{2}}{2 \left(3 x - 5\right)}\, dx = \frac{3 \int \frac{x^{2}}{3 x - 5}\, dx}{2}$$

   1. Rewrite the integrand:

      $$\frac{x^{2}}{3 x - 5} = \frac{x}{3} + \frac{5}{9} + \frac{25}{9 \left(3 x - 5\right)}$$

   2. Integrate term-by-term:

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

         $$\int \frac{x}{3}\, dx = \frac{\int x\, dx}{3}$$

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

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

         So, the result is: $\frac{x^{2}}{6}$

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

         $$\int \frac{5}{9}\, dx = \frac{5 x}{9}$$

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

         $$\int \frac{25}{9 \left(3 x - 5\right)}\, dx = \frac{25 \int \frac{1}{3 x - 5}\, dx}{9}$$

         1. Let $u = 3 x - 5$.

            Then let $du = 3 dx$ and substitute $\frac{du}{3}$:

            $$\int \frac{1}{3 u}\, du$$

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

               $$\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{3}$$

               1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$.

               So, the result is: $\frac{\log{\left(u \right)}}{3}$

            Now substitute $u$ back in:

            $$\frac{\log{\left(3 x - 5 \right)}}{3}$$

         So, the result is: $\frac{25 \log{\left(3 x - 5 \right)}}{27}$

      The result is: $\frac{x^{2}}{6} + \frac{5 x}{9} + \frac{25 \log{\left(3 x - 5 \right)}}{27}$

   So, the result is: $\frac{x^{2}}{4} + \frac{5 x}{6} + \frac{25 \log{\left(3 x - 5 \right)}}{18}$

3. Now simplify:

   $$\frac{x^{2} \log{\left(3 x - 5 \right)}}{2} - \frac{x^{2}}{4} - \frac{5 x}{6} - \frac{25 \log{\left(3 x - 5 \right)}}{18}$$

4. Add the constant of integration:

   $$\frac{x^{2} \log{\left(3 x - 5 \right)}}{2} - \frac{x^{2}}{4} - \frac{5 x}{6} - \frac{25 \log{\left(3 x - 5 \right)}}{18}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{2} \log{\left(3 x - 5 \right)}}{2} - \frac{x^{2}}{4} - \frac{5 x}{6} - \frac{25 \log{\left(3 x - 5 \right)}}{18}+ \mathrm{constant}$$