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NDA 2 2025 Maths, question 86

Topic: Integral calculus

Passage · Q85 to Q86

Let u=∫excos⁡xdxu=\int e^{x}\cos x dx and v=∫exsin⁡xdxv=\int e^{x}\sin x dx.

Q86

Consider the following :

I. dudx=−v\frac{du}{dx}=-v

II. dvdx=−u\frac{dv}{dx}=-u

Which of the above is/are correct?

  1. (a) I only
  2. (b) II only
  3. (c) Both I and II
  4. (d) Neither I nor IICorrect answer

Solution.

Answer: (d) Neither I nor II

  1. By differentiation of the antiderivatives, u′=eˣcos x and v′=eˣsin x.
  2. Neither is generally the negative of the other antiderivative.
  3. For example, with zero constants, at x=0, u′=1 but −v=1/2, and v′=0 but −u=−1/2.
  4. Both claimed identities fail.