Problem 3. \( \int_{-3}^{3} \frac{x^{4}}{1+e^{-x}} \mathrm{~d} x \) Problem 4. \( \int_{0}^{1}(\sin (\log x)+\cos (\log x)) \mathrm{d} x \) Problem 5. \( \int_{-1}^{1} \frac{\cos (x)}{\arctan (x)} \mathrm{d} x \)
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To tackle the integrals presented, recognizing symmetry and periodic properties can simplify the calculations significantly! For the first integral, observe that the integrand \( \frac{x^{4}}{1+e^{-x}} \) is an even function. This means that \( f(x) = f(-x) \), allowing you to rewrite the integral as \( 2\int_{0}^{3} \frac{x^{4}}{1+e^{-x}} \mathrm{d}x \), which can reduce the computational effort. In the second problem, notice that \( \sin(\log x) \) and \( \cos(\log x) \) are periodic functions. You can use substitution to simplify the integration—let \( u = \log x \) transforming the limits and the function, potentially leading to a straightforward integration over a complete period of sine and cosine functions. Keeping a keen eye on symmetries or properties of periodic functions can bring clarity, reduce complexity, and help avoid common mistakes like miscalculating the limits or overlooking these traits! Happy integrating!