Answer
False
Work Step by Step
False.
As a counter example, consider $f(x)=5$.
We have
$\int_a^bf'(x)[f(x)]^4dx=\int_a^b0\cdot 5^4dx=\int_a^b0dx=0$
but
$\frac{1}{5}[f(x)]^5+C=\frac{1}{5}\cdot 5^5+C=625+C\neq 0$
You can help us out by revising, improving and updating this answer.
Update this answerAfter you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.