Discussion
\(\int e^{x}(x^{2} - 2)cos\left(e^{x}(x^{2} - 2x)\right)\) dx=
Answer: A
The integral is \(sin\left(e^{x}(x^{2} - 2x)\right) + C\).
Let \(t = e^{x}(x^{2} - 2x)\). By the product rule:
\[ \begin{aligned} \frac{dt}{dx} &= e^{x}(x^{2} - 2x) + e^{x}(2x - 2) \\ &= e^{x}(x^{2} - 2) \end{aligned} \]
This is exactly the factor outside the cosine, so the integral is \(\int\cos t\,dt = \sin t + C\). The options with \(x^{2} - 2\) inside the sine put the wrong expression inside, and those with an extra \(e^{x}\) or \(x^{2}e^{x}\) in front would differentiate to extra terms.
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