Q216 / 270 · NDA 2021 (Phase 2) · Mathematics

If $f(x)=2^x$, then what is $\int_2^{10} \frac{f^{\prime}(x)}{f(x)} d x$ equal to?

  • A
    $4 \ln 2$
  • B
    $\ln 4$
  • C
    $\ln 5$
  • Correct answer
    D
    $8 \ln 2$

Solution

$\begin{aligned} & \text { Let } f(x)=2^x \\ & \Rightarrow f^{\prime}(x)=2^x \log _e(2) \\ & \text {Now } \int_2^{10} \frac{f^{\prime}(x)}{f(x)} d x=\int_2^{10} \frac{2^x \log _e(2)}{2^x} d x \\ & =\log _e(2)[x]^{10} \\ & =8 \log _e(2)\end{aligned}$

Hence, the correct answer is option: 'D'.

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