Integrating an integral with df(t)?

Question

I'm looking for a technical name or technical term that can be google-searched to find more information.

While working through a problem in Quantum Chemistry, I came across the integral of [1/f(t)] df(t)

The answer, ln f(t) + C

makes sense but this is the first time I've seen the differential df(t).

I can work through it, putting it into more familiar notation by substituting y=f(t) and df(t) = dy.  This results in the expected answer.

I'm a little uneasy with this though--is there a name corresponding to this that I can Google search?

Mark · Accepted Answer

It is just a simplified case of the Stieljes Integral.

See http://mathworld.wolfram.com/StieltjesIntegral.html

(You get horizontal strips because df(t) is the same as dy which is the width of a horizontal strip. )

y = f(t) then dy/dt = f '(t) dt/dt.

Thus dy = f' (t) dt = d(f(t))  which is what the notation means. That is d(f(t)) = f' (t) dt.

Look -- here's a simple example. ∫ x² d(x²). If we let y = x², dy = d(x²) so ∫ y dy = y²/2 = (x²)²/2 = xᶣ/2 + C.

Now let's do it directly. d(x²) = 2x dx. So  ∫ x² d(x²) =  ∫ x² 2x dx = ∫ 2x³ dx = 2(xᶣ/4) = xᶣ/2 + C

Same thing! Don't like x's? Make all x's into t -- now you have a y = f(t) example.

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Integrating an integral with df(t)?

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