Question 1204791
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As I see from the error, made by you in your post, you just know the basic rules
of taking antiderivatives, but are not uniformly attentive.


When you worked on f'(t), you missed the term C*t in this antiderivative.



Taking antiderivatives is a part of Calculus, which you need to learn from your textbook, 
if you want to know the subject.



Taking antiderivative (indefinite integrals) is a key part of Calculus.


To start, look into this introduction


https://www.sparknotes.com/math/calcab/introductiontointegrals/section1/



Normal Calculus student must know the rules of taking antiderivatives as good
as he (or she) knows the multiplication table.



Antiderivative f''(t) in the post by @MathLover1 is INCORRECT.