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NDA 2 2025 Maths, question 97

Topic: Integral calculus

Q97

If I1=∫ee2dxln⁡xI_{1}=\int _{e}^{e^{2}}\frac{dx}{\ln x} and I2=∫12exxdxI_{2}=\int _{1}^{2}\frac{e^{x}}{x} dx, then which one of the following is correct?

  1. (a) I1−I2=0I_{1}-I_{2}=0Correct answer
  2. (b) I1+I2=0I_{1}+I_{2}=0
  3. (c) I1−2I2=0I_{1}-2I_{2}=0
  4. (d) 2I1−I2=02I_{1}-I_{2}=0

Solution.

Answer: (a) I1−I2=0I_{1}-I_{2}=0

  1. In I₁ set t=ln x.
  2. Then x=eᵗ, dx=eᵗdt, and the bounds e,e² become 1,2.
  3. Thus I₁=∫₁²(eᵗ/t)dt=I₂, giving I₁−I₂=0.