How do you use the second fundamental theorem of Calculus to find the derivative of given #int 1/(t^5) dt# from #[1,x]#?

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Answer 1 · 2016-04-07T12:13:39

You could either evaluate the integral and then differentiate or reason as follows:

The Second Fundamental Theorem of Calculus says that

If #f# is continuous on #[a,b]#, then

#int_a^b f(x) dx = F(b)-F(a)#
where #F# is a function for which #F'(x) = f(x)# for all #x# in #[a,b]#.

In this case we are using the variable #t# in the integrand and the variable #x# as the upper limit of integration.

We want the derivative of #int_1^x 1/t^5 dt#.

Note that since we are asked about the interval #[1,x]#, we must have #x > 1# (otherwise the interval is either empty or undefined).

So, #1/t^5# is continuous on #[1,x]#, and

#int_1^x 1/t^5 dt= F(x) - F(1)# where #F# is a function such that #F'(x) = 1/x^5#.

And there is our answer. The derivative of #int_1^x 1/t^5 dt# is #1/x^5#.