Question

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

Answer 1

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)`