Mathematics

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


ANSWER

$$\log 11$$


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
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