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

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Calculate the integral $$ \int \frac{\ln x}{x^{11}} d x $$.
Evaluate the integral by following steps:
- step0: Evaluate using partial integration formula:
  $$\int \frac{\ln{\left(x\right)}}{x^{11}} dx$$
- step1: Evaluate the integral:
  $$\int x^{-11}\ln{\left(x\right)} dx$$
- step2: Prepare for integration by parts:
  $$\begin{align}&u=\ln{\left(x\right)}\&dv=x^{-11}dx\end{align}$$
- step3: Calculate the derivative:
  $$\begin{align}&du=\frac{1}{x}dx\&dv=x^{-11}dx\end{align}$$
- step4: Evaluate the integral:
  $$\begin{align}&du=\frac{1}{x}dx\&v=-\frac{1}{10x^{10}}\end{align}$$
- step5: Substitute the values into formula:
  $$\ln{\left(x\right)}\times \left(-\frac{1}{10x^{10}}\right)-\int \frac{1}{x}\times \left(-\frac{1}{10x^{10}}\right) dx$$
- step6: Calculate:
  $$-\frac{\ln{\left(x\right)}}{10x^{10}}-\int -\frac{1}{10x^{11}} dx$$
- step7: Use properties of integrals:
  $$-\frac{\ln{\left(x\right)}}{10x^{10}}+\int \frac{1}{10x^{11}} dx$$
- step8: Evaluate the integral:
  $$-\frac{\ln{\left(x\right)}}{10x^{10}}-\frac{1}{100x^{10}}$$
- step9: Add the constant of integral C:
  $$-\frac{\ln{\left(x\right)}}{10x^{10}}-\frac{1}{100x^{10}} + C, C \in \mathbb{R}$$
The integral of $$ \frac{\ln x}{x^{11}} $$ with respect to $$ x $$ is given by:

$$ \int \frac{\ln x}{x^{11}} dx = -\frac{\ln(x)}{10x^{10}} - \frac{1}{100x^{10}} + C $$

where $$ C $$ is the constant of integration.
The integral of $$ \frac{\ln x}{x^{11}} $$ with respect to $$ x $$ is $$ -\frac{\ln(x)}{10x^{10}} - \frac{1}{100x^{10}} + C $$.

Evaluate the following integral using integration by parts.

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