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Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. #int_(e^x)^0 2sin^(2)t dt# ?

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Jun 24, 2018

From FTC part 1:

#d/dx int_a^x f(t) dt = f(x) implies d/dx int_x^a f(t) dt = - f(x)#

And applying chain rule:

# d/dx int_(g(x))^a f(t) dt = - f(g(x)) * g'(x)#

So:

#d/dx int_(e^x)^0 2sin^(2)t \ dt#

#= - 2sin^(2)(e^x) (d (e^x))/dx#

#= - 2e^xsin^(2)(e^x) #

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