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NDA 2025 Maths · Mathematics · Integrals · easy

Using the substitution u=1+sin x, the value of ∫0^π/2(cos x)/(1+sin x) dx is

Single correct option

Using the substitution u=1+sin⁡xu=1+\sin x, the value of ∫0π/2cos⁡x1+sin⁡x dx\displaystyle\int_0^{\pi/2}\dfrac{\cos x}{1+\sin x}\,dx is
  1. A. ln⁡12\ln\dfrac12
  2. B. π4\dfrac\pi4
  3. C. 11
  4. D. ln⁡2\ln2
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Answer: (D) ln⁡2\ln2

The numerator is the derivative of the denominator, so the integral is ln⁡(1+sin⁡x)\ln(1+\sin x). Evaluating, ln⁡2−ln⁡1=ln⁡2\ln2-\ln1=\ln2. The option ln⁡12\ln\frac12 reverses the limits, π4\frac\pi4 would belong to an inverse tangent, and 11 is ∫0π/2cos⁡x dx\int_0^{\pi/2}\cos x\,dx, which ignores the denominator.
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Original question written in the NDA 2025 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake