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NDA 1 2025 Maths, question 82

Topic: Integral calculus

Passage · Q81 to Q82

Let the function y=(1−cos⁡x)−1y = \left( 1 - \cos x \right)^{- 1}, where x≠2nπx \neq 2n\pi and nn is an integer.

Q82

What is ∫y dx\int y\, dx equal to? Where cc is the constant of integration.

  1. (a) −tan⁡(x/2)+c- \tan(x/2) + c
  2. (b) −cot⁡(x/2)+c- \cot(x/2) + cCorrect answer
  3. (c) tan⁡(x/2)+c\tan(x/2) + c
  4. (d) cot⁡(x/2)+c\cot(x/2) + c

Solution.

Answer: (b) −cot⁡(x/2)+c- \cot(x/2) + c

  1. Use 1−cos⁡x=2sin⁡2(x/2)1 - \cos x = 2\sin^{2}(x/2).
  2. Then ∫y dx=12∫csc⁡2(x/2) dx=−cot⁡(x/2)+c\int y\, dx = \frac{1}{2}\int\csc^{2}(x/2)\, dx = - \cot(x/2) + c.