Mathematics

# Let $f(0) = 0$ and $\displaystyle \int_{0}^{2}{f}'(2t)e^{f(2t)} \:dt=5$.Then the value of $f (4)$ is?

$\log 11$

##### SOLUTION
We have $\displaystyle \int_{0}^{2 }{f}'(2t)e^{f(2t)}dt=5$

Substitute $e^{f(2t)}=y$

$\therefore 2{f}'(2t)e^{f(2t)}dt=dy$

$\therefore \dfrac{1}{2} \displaystyle \int_{e^{f(0)}}^{e^{f(4)}} dy=5$

or $\displaystyle \int_{e^{f(0)}}^{e^{f(4)}} dy=10$

or $e^{f(4)}-e^{f(0)}=10$

or $e^{f(4)}=10+1=11$

or $f(4)=\log 11$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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