How do you use the Fundamental Theorem of Calculus to find the derivative of #int sqrt(1+ sec(t)) dt# from x to pi?

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Answer 1 · 2016-07-02T16:00:22

#= - sqrt(1+ sec(x))#

you're interested in the second part of the Fundamental Theorem which states that

if #F(x) = int_a^x \ f(t) \ dt#

then #(dF)/dx = f(x) #

here you want

#d/dx int_x^pi \ sqrt(1+ sec(t)) \ dt#

to get that into the form required by the FTC pt 2, and using just the summation signs to save time, we observe that

#int_x^pi = - int_color{red}{pi}^color{red}{x} #

so if we are to use the FTC part 2, we re-write it as

#d/dx (int_x^pi \ sqrt(1+ sec(t)) \ dt)#

#d/dx (-int_color{red}{pi}^color{red}{x} \ sqrt(1+ sec(t)) \dt )#

#-d/dx (int_pi^x \ sqrt(1+ sec(t)) \ dt )#

#= - sqrt(1+ sec(x))#