We know if g is continuous on ( a , b ) and F (...

Yahir Tucker

Yahir Tucker

Answered

2022-06-24

We know if g is continuous on ( a , b ) and F ( x ) = a x g ( t ) d t, then
F ( x ) = g ( x )
But, how about if we have
F ( x ) = a h ( x ) g ( t ) d t
What should F ( x ) be?? can we still apply fundamental theorem of calculus?

Answer & Explanation

sleuteleni7

sleuteleni7

Expert

2022-06-25Added 28 answers

In that case you have :
a h ( x ) g ( t ) d t = G ( h ( x ) ) G ( a )
where G ( x ) = g ( x ). Hence :
[ G ( h ( x ) ) ] = h ( x ) G ( h ( x ) ) = h ( x ) g ( h ( x ) )
by the chain rule.
telegrafyx

telegrafyx

Expert

2022-06-26Added 8 answers

Yes, but you need also the chain rule. Obviously, h must be differentiable.
F ( x ) = h ( x ) g ( h ( x ) )

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