I'm having some trouble with part of a problem from Apostol Volume 1(Section 6.26, Number 6). For completeness I'll include the whole question:
A function $F$ is defined by the following indefinite integral: $$ F(x) = \int_1^x \frac{e^t}{t}\cdot dt,\,\,\, x > 0 $$ (a) For what values of $x$ is it true that $\log\,\, x \leq F(x)$ ?
(b) Prove that $$ \int_1^x\frac{e^t}{t+a}\cdot dt = e^{-a}[F(x+a) - F(1+a)] $$
(c) In a similar way, express the following integrals in terms of $F$: $$ \int_1^x\frac{e^{at}}{t}\cdot dt, \int_1^x\frac{e^t}{t^2}\cdot dt, \int_1^xe^{\frac{1}{t}}\cdot dt $$ I have successfully solved parts (a), (b) and the first two parts of (c). I seem to be having trouble with the last part of (c), I managed to solve part (b) and the first two parts of (c) by relying on the basic properties of the integral (linearity, invariance, additivity, expansion of the interval of integration, comparison) but for the last integral of part (c) I have not been able to come up with anything at all.
All I have tried is starting with the answer and working backwards in order to establish a property but this was unsuccessful for me.
NB: the answer in the back of the book for the last part of (c) is $$ x\cdot e^{\frac{1}{x}} - e - F(\frac{1}{x}) $$ All I am interested in is some hints, not the answer.
Let $u = 1/t$, then $du = -\dfrac{dt}{t^2}$, then $I = -\displaystyle \int_{1}^{1/x} \dfrac{e^u}{u^2}du = \displaystyle \int_{1}^{1/x} e^ud\left(1/u\right) = x\cdot e^{1/x} - e - F\left(1/x\right)$ by integration by parts.