How do you use the second fundamental theorem of Calculus to find the derivative of given #int tan(t^4)-1) dt# from #[1, x^3]#?

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Answer 1 · 2016-05-03T14:07:43

Please see the explanation section below.

The second fundamental theorem of calculus tells us that

#g(u) = int_1^u (tan(t^4)-1) dt = F(u)-F(1)#

where #F# is an antiderivative of #(tan(t^4)-1)#.

That is, #F'(u) = tan(u^4)-1#.

Nore that, since #$f(1)# is simply some constant, we have #g'(u) = F'(u) = tan(u^4)-1#.

We have been asked for the derivative of #g(x^3)#, so we'll use the chain rule.

#d/dx(g(x^3)) = g'(x^3)*d/dx(x^3) = (tan((x^3)^4)-1)d/dx(x^3)#

# = 3x^2(tan(x^12)-1)#