Evaluate , and hence prove that…

MATHEMATICS · Subjective · AP · Calculus BC · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Evaluate

\int x^2 \log(1-x^2)\,dx

, and hence prove that

\frac{1}{1.5} + \frac{1}{2.7} + \frac{1}{3.9} + \dots = \frac{2}{3} \log 2 - \frac{8}{9}