Integral of x*ln(2x-3) dx

1. Use integration by parts:

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

   Let $u{\left(x \right)} = \log{\left(2 x - 3 \right)}$ and let $\operatorname{dv}{\left(x \right)} = x$.

   Then $\operatorname{du}{\left(x \right)} = \frac{2}{2 x - 3}$.

   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. Rewrite the integrand:

   $$\frac{x^{2}}{2 x - 3} = \frac{x}{2} + \frac{3}{4} + \frac{9}{4 \left(2 x - 3\right)}$$

3. Integrate term-by-term:

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

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

      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: $\frac{x^{2}}{4}$

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

      $$\int \frac{3}{4}\, dx = \frac{3 x}{4}$$

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

      $$\int \frac{9}{4 \left(2 x - 3\right)}\, dx = \frac{9 \int \frac{1}{2 x - 3}\, dx}{4}$$

      1. Let $u = 2 x - 3$.

         Then let $du = 2 dx$ and substitute $\frac{du}{2}$:

         $$\int \frac{1}{2 u}\, du$$

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

            $$\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{2}$$

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

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

         Now substitute $u$ back in:

         $$\frac{\log{\left(2 x - 3 \right)}}{2}$$

      So, the result is: $\frac{9 \log{\left(2 x - 3 \right)}}{8}$

   The result is: $\frac{x^{2}}{4} + \frac{3 x}{4} + \frac{9 \log{\left(2 x - 3 \right)}}{8}$

4. Now simplify:

   $$\frac{x^{2} \log{\left(2 x - 3 \right)}}{2} - \frac{x^{2}}{4} - \frac{3 x}{4} - \frac{9 \log{\left(2 x - 3 \right)}}{8}$$

5. Add the constant of integration:

   $$\frac{x^{2} \log{\left(2 x - 3 \right)}}{2} - \frac{x^{2}}{4} - \frac{3 x}{4} - \frac{9 \log{\left(2 x - 3 \right)}}{8}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{2} \log{\left(2 x - 3 \right)}}{2} - \frac{x^{2}}{4} - \frac{3 x}{4} - \frac{9 \log{\left(2 x - 3 \right)}}{8}+ \mathrm{constant}$$