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NDA 2025
Exam
NDA
Year
2025
Subject
Mathematics
Chapter
Definite Integration

NDA 2025 · Mathematics · Definite Integration

If I_1 = ∫_e^ e^2 (dx)/( x) and I_2 = ∫_1^2 (e^x)/(x)\, dx , then...

If I1=∫ee2dxln⁡xI_1 = \int_e^{e^2} \dfrac{dx}{\ln x} and I2=∫12exx dxI_2 = \int_1^2 \dfrac{e^x}{x}\, dx, then which one of the following is correct?

Options

  1. AI1−I2=0I_1 - I_2 = 0
  2. BI1+I2=0I_1 + I_2 = 0
  3. CI1−2I2=0I_1 - 2I_2 = 0
  4. D2I1−I2=02I_1 - I_2 = 0

Correct answer

A. I1−I2=0I_1 - I_2 = 0

Step-by-step solution

I1=∫ee2dxln⁡xI_1 = \int_e^{e^2} \dfrac{dx}{\ln x}

Substituting ln⁡x=t⇒x=et⇒dx=etdt\ln x = t \Rightarrow x = e^t \Rightarrow dx = e^t dt

When x=ex = e, t=1t = 1

When x=e2x = e^2, t=2t = 2

I1=∫12ettdtI_1 = \int_1^2 \dfrac{e^t}{t} dt

Since the definite integral is independent of the variable of integration,

I1=∫12exxdx=I2I_1 = \int_1^2 \dfrac{e^x}{x} dx = I_2

I1−I2=0I_1 - I_2 = 0

Answer: I1−I2=0I_1 - I_2 = 0
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