What is u+v equal to?
For the following two (02) items: Let u=\int e^{x}\cos x\,dx and v=\int e^{x}\sin x\,dx.
- A. -\frac{du}{dx}
- B. -\frac{dv}{dx}
- C. \frac{du}{dx}
- D. \frac{dv}{dx} ✓
Correct Answer: D. \frac{dv}{dx}
Explanation
By the Fundamental Theorem of Calculus, \frac{du}{dx} = e^x \cos x and \frac{dv}{dx} = e^x \sin x. Standard integration gives u = \frac{e^x}{2}(\cos x + \sin x) and v = \frac{e^x}{2}(\sin x - \cos x). Adding them yields u+v = e^x \sin x, which is precisely \frac{dv}{dx}.
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