Question

Let $f:R\to R$ be a differentiable function and $f\left(1\right)=4,$then the value of $li{m}_{x\to 1}{\int }_4^{f\left(x\right)}\frac{2t}{(x-1)}dt$ is:

• A. $8f\text{'}\left(1\right)$
• B. $4f\text{'}\left(1\right)$
• C. $2f\text{'}\left(1\right)$
• D. $f\text{'}\left(1\right)$

Answer 1

The correct option is A

$8f\text{'}\left(1\right)$

Solving the integral:

$$\begin{gathered} I=li{m}_{x\to 1}{\int }_4^{f\left(x\right)}\frac{2t}{(x-1)}dt \\ =li{m}_{x\to 1}{\left[\frac{t^2}{x-1}\right]}_4^{f\left(x\right)} \\ =li{m}_{x\to 1}\left[\frac{{\left|f\left(x\right)\right|}^2-16}{1}\right]\left[\frac{0}{0}\text{form, Applying L's Hsopital rule}\right] \\ =li{m}_{x\to 1}\frac{2f\left(x\right).f\text{'}\left(x\right)}{1} \\ =2f\left(1\right)f\text{'}\left(1\right) \\ =8f\text{'}\left(1\right) \end{gathered}$$

Therefore, the correct answer is Option (A).