Question:

Step 1: Evaluate the given integral using integration by parts framework. To solve the integral $I = \int (x^4 + 2x^2 + 1)\sin x\, dx$, we will use repeated integration by parts. The standard result for $\int P(x)\sin x\, dx$ gives: $$\int P(x)\sin x\, dx = -P(x)\cos x + P'(x)\sin x + P''(x)\cos x - P'''(x)\sin x - \cdots$$ Let $P(x) = x^4 + 2x^2 + 1$. Then we find the derivatives of $P(x)$. Step 2: Compute the derivatives of $P(x)$. We have: - $P(x) = x^4 + 2x^2 + 1$ - $P'(x) = 4x^3 + 4x$ - $P''(x) = 12x^2 + 4$ - $P'''(x) = 24x$ - $P^{(4)}(x) = 24$ - $P^{(5)}(x) = 0$ Using these derivatives, we can express the integral in terms of $P(x)$ and its derivatives. Step 3: Express the integral in terms of $P(x)$ and its derivatives. The formula for the integral becomes: $$\int P(x)\sin x\, dx = \left[-P(x) + P''(x) - P^{(4)}(x)\right]\cos x + \left[P'(x) - P'''(x) + P^{(5)}(x)\right]\sin x + C$$ Now, we can substitute the derivatives of $P(x)$ into this formula. Step 4: Substitute the derivatives of $P(x)$ into the formula. Substituting the derivatives into the formula gives: $$\int (x^4 + 2x^2 + 1)\sin x\, dx = \left[-(x^4 + 2x^2 + 1) + (12x^2 + 4) - 24\right]\cos x + \left[(4x^3 + 4x) - 24x + 0\right]\sin x + C$$ Simplifying the expressions inside the brackets, we get: $$\int (x^4 + 2x^2 + 1)\sin x\, dx = \left[-x^4 - 2x^2 - 1 + 12x^2 + 4 - 24\right]\cos x + \left[4x^3 + 4x - 24x\right]\sin x + C$$ $$\int (x^4 + 2x^2 + 1)\sin x\, dx = \left[-x^4 + 10x^2 - 21\right]\cos x + \left[4x^3 - 20x\right]\sin x + C$$ Step 5: Identify $f(x)$ and $g(x)$. From the simplified expression, we can identify: - $g(x) = -x^4 + 10x^2 - 21$ - $f(x) = 4x^3 - 20x$ These are the coefficients of $\cos x$ and $\sin x$ respectively. Step 6: Analyze the properties of $f(x)$ and $g(x)$. $f(x)$ is a cubic polynomial (odd degree), so it is an onto function from $\mathbb{R}$ to $\mathbb{R}$. $f'(x) = 12x^2 - 20 = 0$ at $x = \pm\sqrt{5/3}$, so $f$ has a local maximum and minimum, meaning it is NOT one-one. $g(x)$ is a degree-4 polynomial (even degree) with a range of $(-\infty, 4]$, which is a proper subset of $\mathbb{R}$, making it into. Also, $g$ is even, so $g(-x) = g(x)$, making it many-one. Step 7: Determine the number of non-derivable points of $|f(x)|$ and $|g(x)|$. $|f(x)|$ is non-derivable where $f(x) = 0$ and $f$ changes sign, which occurs at $x = 0, \pm\sqrt{5}$. So, $|f(x)|$ is non-derivable at 3 points. $g(x) = 0$ at $x = \pm\sqrt{3}, \pm\sqrt{7}$, and at each of these points, $g$ changes sign. So, $|g(x)|$ is non-derivable at 4 points. Step 8: Conclude the final answer based on the analysis of $f(x)$ and $g(x)$. Based on the analysis, $f(x)$ matches options (2) and (3), $g(x)$ matches options (2) and (4), $|f(x)|$ matches option (5), and $|g(x)|$ matches option (6). Therefore, the correct answer is option (1).