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.)

To evaluate the integral \( \int_{0}^{1} \frac{x^{10}}{x^{11}-1} dx \), we can first recognize that the integrand has a singularity at \( x = 1 \). To analyze this, we can split the integral and consider the limit: \[ \int_{0}^{1} \frac{x^{10}}{x^{11}-1} dx = \lim_{b \to 1^-} \int_{0}^{b} \frac{x^{10}}{x^{11}-1} dx \] Now, to simplify the integrand, notice that as \( x \) approaches 1, \( x^{11} - 1 \) approaches 0. A helpful substitution is to let \( x^{11} = t \), giving \( dx = \frac{1}{11} t^{\frac{10}{11}} dt \). The limits change from \( x = 0 \) to \( x = 1 \), mapping \( t \) from \( 0 \) to \( 1 \). The integral becomes: \[ \int_{0}^{1} \frac{t^{10/11}}{t - 1} \frac{1}{11} dt \] Now, as we evaluate the limit, we can use the theory of improper integrals. The integrand has a logarithmic divergence at \( t = 1 \), indicating the integral will diverge. Thus, we state that the improper integral diverges. Therefore, the conclusion is: The improper integral diverges.