Can you apply the fundamental theorem of calculus with the variable inside the integrand? For examp

Leland Morrow 2022-07-01 Answered
Can you apply the fundamental theorem of calculus with the variable inside the integrand?
For example: d d x ( a x x f ( t ) d t )
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Answers (1)

hopeloothab9m
Answered 2022-07-02 Author has 25 answers
By definition
d d x a x f ( x , t ) d t = lim Δ x 0 1 Δ x [ a x + Δ x f ( x + Δ x , t ) d t a x f ( x , t ) d t ]
= lim Δ x 0 1 Δ x [ a x + Δ x ( f ( x , t ) + f x Δ x ) d t a x f ( x , t ) d t ]
= lim Δ x 0 1 Δ x [ a x + Δ x f ( x , t ) d t a x f ( x , t ) d t + a x + Δ x f x Δ x d t ]
= lim Δ x 0 1 Δ x [ x x + Δ x f ( x , t ) d t + a x + Δ x f x Δ x d t ]
= f ( x , x ) + lim Δ x 0 a x + Δ x f x d t
= f ( x , x ) + a x x f ( x , t ) d t
In conclusion, you will only get the first term if you apply the FTOC blindly. Because the integrand contains x, you will have the second extra term.
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