How do you use the second fundamental theorem of Calculus to find the derivative of given #int ((sin^3)(t))dt# from #[0, e^x]#?

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Answer 1 · 2016-07-09T19:34:11

#e^x sin^3 e^x #

you want

#d/dx int_0^(e^x) sin^3(t) \ dt#

by the FTC we know that

#d/(du) int_a^u f(t) \ dt = f(u)#

and if #u = u(x)# we also know that

#d/(dx) int_a^u f(t) \ dt = f(u)* (du)/dx#

we apply that here with #f(t) = sin^3 t#, #a = 0# and #u = e^x# to get

#sin^3 e^x * e^x#