If $f(2-x) = f(2+x)$ and $f(4-x) = f(4+x)$ for all $x$ and $f(x)$ is a function for which $\int_0^2 f(x)dx = 5$, then $\int_0^{50} f(x)dx$ is equal to:
Step-by-Step Solution
Key Concept: Replace variable in series, multiply by $x^2$, integrate term-by-term, and use integration by parts with logarithmic functions.
Starting with the series $\ln(1-x) = -\left(x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \ldots\right)$, replace $x$ with $x^2$ to get $\ln(1-x^2) = -\left(x^2 + \frac{x^4}{2} + \frac{x^6}{3} + \frac{x^8}{4} + \ldots\right)$. Multiply by $x^2$ and integrate from 0 to 1 using integration by parts. The limit term vanishes as $x^3 - 1 \to 0$, yielding $\frac{1}{1.5} + \frac{1}{2.7} + \frac{1}{3.9} + \ldots = \frac{2}{3}\ln 2 - \frac{8}{9}$.
Correct Answer: 2,3