Evaluate the following integral using integration by parts. \[ \int x^{11} \ln x d x \]

Calculate the integral \( \int x^{11} \ln x d x \). Evaluate the integral by following steps: - step0: Evaluate using partial integration formula: \(\int x^{11}\ln{\left(x\right)} dx\) - step1: Prepare for integration by parts: \(\begin{align}&u=\ln{\left(x\right)}\\&dv=x^{11}dx\end{align}\) - step2: Calculate the derivative: \(\begin{align}&du=\frac{1}{x}dx\\&dv=x^{11}dx\end{align}\) - step3: Evaluate the integral: \(\begin{align}&du=\frac{1}{x}dx\\&v=\frac{x^{12}}{12}\end{align}\) - step4: Substitute the values into formula: \(\ln{\left(x\right)}\times \frac{x^{12}}{12}-\int \frac{1}{x}\times \frac{x^{12}}{12} dx\) - step5: Calculate: \(\frac{x^{12}\ln{\left(x\right)}}{12}-\int \frac{x^{11}}{12} dx\) - step6: Evaluate the integral: \(\frac{x^{12}\ln{\left(x\right)}}{12}-\frac{x^{12}}{144}\) - step7: Simplify the expression: \(\frac{1}{12}x^{12}\ln{\left(x\right)}-\frac{x^{12}}{144}\) - step8: Add the constant of integral C: \(\frac{1}{12}x^{12}\ln{\left(x\right)}-\frac{x^{12}}{144} + C, C \in \mathbb{R}\) The integral of \(x^{11} \ln x\) with respect to \(x\) is \(\frac{1}{12}x^{12}\ln(x) - \frac{x^{12}}{144} + C\), where \(C\) is an arbitrary constant.