What is \int y\,dx equal to?
Consider the following for the two (02) items that follow : Let the function y=(1-\cos x)^{-1}, where x \neq 2n\pi and n is an integer.
- A. -\tan(x/2)+c
- B. -\cot(x/2)+c ✓
- C. \tan(x/2)+c
- D. \cot(x/2)+c
Correct Answer: B. -\cot(x/2)+c
Explanation
Using the half-angle formula, 1-\cos x = 2\sin^2(x/2). The integral becomes \int \frac{1}{2\sin^2(x/2)}\,dx = \frac{1}{2}\int \csc^2(x/2)\,dx = \frac{1}{2} \frac{-\cot(x/2)}{1/2} + c = -\cot(x/2) + c.
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