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Evaluate the integral $\int_0^1 x^2 dx$.

Mathematical analysis is a branch of mathematics that deals with the study of limits, sequences, series, and calculus. This paper provides an overview of the key concepts and techniques in mathematical analysis, with a focus on solutions to selected problems. We draw on the textbook "Mathematical Analysis" by Vladimir Zorich as a primary reference.

As $x$ approaches 0, $f(g(x))$ approaches 1.

We have $f(g(x)) = f(\frac11+x) = \frac1\frac11+x = 1+x$.

Using the product rule, we have $f'(x) = 2x \sin x + x^2 \cos x$.