Evaluate the following integral or state that it diverges. Select the correct choice and, if necessary, fill in the answer box to complete your choice. A. $$ \frac{x^{10}}{x^{11}-1} d x $$ The improper integral converges and $$ \int_{0}^{1} \frac{x^{10}}{x^{11}-1} d x=\square $$. (Type an exact answer, using $$ \pi $$ as needed.)

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Calculate the integral $$ \int \frac{x^{10}}{x^{11}-1} dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \frac{x^{10}}{x^{11}-1} dx$$
- step1: Use the substitution $$dx=\frac{1}{11x^{10}} dt$$ to transform the integral$$:$$
  $$\int \frac{x^{10}}{x^{11}-1}	imes \frac{1}{11x^{10}} dt$$
- step2: Simplify:
  $$\int \frac{1}{11x^{11}-11} dt$$
- step3: Use the substitution $$t=x^{11}$$ to transform the integral$$:$$
  $$\int \frac{1}{11t-11} dt$$
- step4: Rewrite the expression:
  $$\int \frac{1}{11}	imes \frac{1}{t-1} dt$$
- step5: Use properties of integrals:
  $$\frac{1}{11}	imes \int \frac{1}{t-1} dt$$
- step6: Evaluate the integral:
  $$\frac{1}{11}\ln{\left(\left|t-1ight|ight)}$$
- step7: Substitute back:
  $$\frac{1}{11}\ln{\left(\left|x^{11}-1ight|ight)}$$
- step8: Add the constant of integral C:
  $$\frac{1}{11}\ln{\left(\left|x^{11}-1ight|ight)} + C, C \in \mathbb{R}$$
The integral of $$ \frac{x^{10}}{x^{11}-1} $$ with respect to $$ x $$ is $$ \frac{1}{11} \ln(|x^{11}-1|) + C $$, where $$ C $$ is an arbitrary constant.

Now, we need to evaluate the improper integral from 0 to 1.

To evaluate the improper integral, we need to find the limit as the upper bound approaches 1.

Let's calculate the limit:
Evaluate the limit by following steps:
- step0: Evaluate the limit:
  $$\lim _{xightarrow 1}\left(\frac{1}{11}\ln{\left(\left|x^{11}-1ight|ight)}ight)$$
- step1: Rewrite the expression:
  $$\frac{1}{11}	imes \lim _{xightarrow 1}\left(\ln{\left(\left|x^{11}-1ight|ight)}ight)$$
- step2: Calculate:
  $$\frac{1}{11}\left(-\inftyight)$$
- step3: Simplify:
  $$-\infty$$
The limit as $$ x $$ approaches 1 is $$ -\infty $$, which means the improper integral diverges.

Therefore, the correct answer is:

B. The improper integral diverges.
The improper integral $$ \int_{0}^{1} \frac{x^{10}}{x^{11}-1} dx $$ diverges.

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