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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Definite Integration

NDA 2021 · Mathematics · Definite Integration

If f(x)=2^x , then what is ∫_2^ 10 f^ (x) f(x) d x equal to?

If f(x)=2xf(x)=2^x, then what is ∫210f′(x)f(x)dx\int_2^{10} \frac{f^{\prime}(x)}{f(x)} d x equal to?

Options

  1. A4ln⁡24 \ln 2
  2. Bln⁡4\ln 4
  3. Cln⁡5\ln 5
  4. D8ln⁡28 \ln 2

Correct answer

D. 8ln⁡28 \ln 2

Step-by-step solution

 Let f(x)=2x⇒f′(x)=2xlog⁡e(2)Now ∫210f′(x)f(x)dx=∫2102xlog⁡e(2)2xdx=log⁡e(2)[x]10=8log⁡e(2)\begin{aligned} & \text{ Let } f(x)=2^x \\ & \Rightarrow f^{\prime}(x)=2^x \log _e(2) \\ & \text{Now } \int_2^{10} \frac{f^{\prime}(x)}{f(x)} d x=\int_2^{10} \frac{2^x \log _e(2)}{2^x} d x \\ & =\log _e(2)[x]^{10} \\ & =8 \log _e(2)\end{aligned}
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