Integral of x×ln(x+1) dx

1. Use integration by parts:

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

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

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

   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:

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

   1. Rewrite the integrand:

      $$\frac{x^{2}}{x + 1} = x - 1 + \frac{1}{x + 1}$$

   2. Integrate term-by-term:

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

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

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

         $$\int \left(-1\right)\, dx = - x$$

      1. Let $u = x + 1$.

         Then let $du = dx$ and substitute $du$:

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

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

         Now substitute $u$ back in:

         $$\log{\left(x + 1 \right)}$$

      The result is: $\frac{x^{2}}{2} - x + \log{\left(x + 1 \right)}$

   So, the result is: $\frac{x^{2}}{4} - \frac{x}{2} + \frac{\log{\left(x + 1 \right)}}{2}$

3. Now simplify:

   $$\frac{x^{2} \log{\left(x + 1 \right)}}{2} - \frac{x^{2}}{4} + \frac{x}{2} - \frac{\log{\left(x + 1 \right)}}{2}$$

4. Add the constant of integration:

   $$\frac{x^{2} \log{\left(x + 1 \right)}}{2} - \frac{x^{2}}{4} + \frac{x}{2} - \frac{\log{\left(x + 1 \right)}}{2}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{2} \log{\left(x + 1 \right)}}{2} - \frac{x^{2}}{4} + \frac{x}{2} - \frac{\log{\left(x + 1 \right)}}{2}+ \mathrm{constant}$$