The heckler is confusing the first and second fundamental theorems of calculus.

(1) The first fundamental theorem states that if `f ` is continuous on a closed interval `[a,b] ` and `F ` is an antiderivative of `f ` on the interval `[a,b] ` then `int_a^b f(x)dx=F(b)-F(a) ` .

In his example, let `F(x)=g(x) ` . Then, with `F(x)=sinx+5,f(x)=cosx ` then`int_a^x costdt=sinx-sina ` .

(2) The second fundamental theorem states that if `f ` is continuous on an open interval ` I ` containing `a ` , then for every `x ` in the interval ` `` ``d/(dx) [ int_a^x f(t)dt]=f(x) ` . From his example this implies that ` d/(dx)[int_a^x costdt]=cosx ` .

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In neither case is there an implication that `int_a^x f(t)dt=F(x) ` , or in terms of the example `int_a^x costdt=sinx+5 ` .