A formula for ddx∫g(x)h(x)ftdt
The Fundamental Theorem of Calculus: Suppose f is continuous on [a, b], then
∫abfxdx=Fb-F(a), where F is antiderivative of f, that is F'=f.
If Ft is the antiderivative of f(t) then by Fundamental Theorem of Calculus ∫g(x)h(x)ftdt=Fhx-F(gx)
By taking derivative on both sides, the above equation becomes
By applying the chain rule, theabove equation becomes
Substituting F'=f in the above equation, it becomes
Hence, the formula for ddx∫g(x)h(x)ftdt is