# Need help simplifying integral using integration by parts rule

$$Y(\tau)=\int_{0}^{\tau} t\frac{dF}{dt}dt \:\:\:\:\:\: ; \:\:\: Y(0)=0$$ $$X(\tau)=F(\tau) \:\:\:\:\:\: ; \:\:\: X(0)=0$$

$$Q(X(\tau))=\int_{X(0)}^{X(\tau)} Y(t)dX$$

First real quick: Can I get away with $\int t \: dF=tF(t)$? (Pretty sure not)

Assuming the answer to this is no, I am trying to avoid an expression for $Q(X(t))$ that involves twice integrating, or any integration in general that will result in a huge complicated equation. For example, I prefer to deal with $\int F \: dt$ rather than $\int t \: dF$.

So, is it possible to use integration by parts to express $Q(X(\tau))$ in the following way:

$$Y(\tau)=\tau F(\tau)-\int_{0}^{\tau} F(t)dt$$

$$Q(X(\tau))=Y(\tau)X(\tau)-\int_{X(0)}^{X(\tau)} \frac{dY}{dX}dX$$

$$=Y(\tau)(X(\tau)-1)$$

Substituing in for $Y(\tau)$ and $X(\tau)=F(\tau)$:

$$Q(X(\tau))=(\tau F(\tau)-\int_{0}^{\tau} F(t)dt)(F(\tau)-1)$$

Is that allowed? I'm open to suggestions for better expressions.

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It's rather confusing to use the same letter for a free variable and a bound one. – joriki Jan 20 '13 at 17:15
@joriki Duly edited. How about now? – ben Jan 20 '13 at 21:49
It got worse. You're still using $t$ both for a free variable and a bound one, except now you're not even mentioning the free variable in the integral anymore and have turned it into an indefinite integral instead; so instead of resolving the variable name confusion you've created a new problem in that it doesn't make sense to equate a function to an indefinite integral (which is only defined up to an additive constant). What you need is e.g. $$Y(\tau)=\int_0^\tau t\frac{\mathrm dF}{\mathrm dt}\mathrm dt\;.$$ – joriki Jan 20 '13 at 22:24
@joriki Duly edited. How about now? – ben Jan 21 '13 at 2:30
Yes, it looks OK now. – joriki Jan 21 '13 at 8:44

No, this isn't right – too bad, you would have invented a generic method for solving all integrals :-) If

$$\def\quest{\stackrel?=} \int_{X(0)}^{X(\tau)}Y(t)\,\mathrm dX\quest Y(\tau)(X(\tau)-1)$$

were correct, we could solve all integrals if we know a point $x_0$ where the integrand vanishes:

\begin{align} \int_a^bf(x)\,\mathrm dx&=\int_{x_0}^bf(x)\,\mathrm dx-\int_{x_0}^af(x)\,\mathrm dx\\ &\quest f(b)(b-1)-f(a)(a-1)\;. \end{align}

The error is in the first step,

$$\int_{X(0)}^{X(\tau)}Y(t)\,\mathrm dX\quest Y(\tau)X(\tau)-\int_{X(0)}^{X(\tau)}\frac{\mathrm dY}{\mathrm dX}\mathrm dX\;.$$

That's not a valid integration by parts. A valid integration by parts would look like this

$$\int Y\mathrm dX=YX-\int X\mathrm dY$$

or like this if you prefer (with the same result):

$$\int Y\mathrm dX=\int Y\frac{\mathrm dX}{\mathrm dt}\mathrm dt=YX-\int X\frac{\mathrm dY}{\mathrm dt}\mathrm dt=YX-\int X\mathrm dY\;.$$

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