Consider the following: I. \frac{du}{dx}=-v II. \frac{dv}{dx}=-u Which of the above is/are correct?
For the following two (02) items: Let u=\int e^{x}\cos x\,dx and v=\int e^{x}\sin x\,dx.
- A. I only
- B. II only
- C. Both I and II
- D. Neither I nor II ✓
Correct Answer: D. Neither I nor II
Explanation
By definition of the given integrals, taking the derivative with respect to x yields \frac{du}{dx} = e^x \cos x and \frac{dv}{dx} = e^x \sin x. Since u = \frac{e^x}{2}(\cos x + \sin x) and v = \frac{e^x}{2}(\sin x - \cos x), it is clear that \frac{du}{dx} \neq -v and \frac{dv}{dx} \neq -u. Thus, neither statement is correct.
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