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NDA 2 2025 Maths, question 95

Topic: Integral calculus

Q95

What is ∫dx10x+10−x\int \frac{dx}{10^{x}+10^{-x}} equal to?

  1. (a) tan⁡−1(10x)+c\tan ^{-1}(10^{x})+c
  2. (b) (ln⁡10)tan⁡−1(10x)+c(\ln 10)\tan ^{-1}(10^{x})+c
  3. (c) 1ln⁡10tan⁡−1(10x)+c\frac{1}{\ln 10}\tan ^{-1}(10^{x})+cCorrect answer
  4. (d) ln⁡(10x+10−x)+c\ln (10^{x}+10^{-x})+c

Solution.

Answer: (c) 1ln⁡10tan⁡−1(10x)+c\frac{1}{\ln 10}\tan ^{-1}(10^{x})+c

  1. Multiply numerator and denominator by 10ˣ, then set t=10ˣ so dt=(ln10)10ˣdx.
  2. The integral becomes (1/ln10)∫dt/(1+t²)=(1/ln10)tan⁻¹(10ˣ)+C.
  3. Differentiation restores the original integrand.