Paper 1, Section I, E

(a) Let $f$ be continuous in $[a, b]$, and let $g$ be strictly monotonic in $[\alpha, \beta]$, with a continuous derivative there, and suppose that $a=g(\alpha)$ and $b=g(\beta)$. Prove that

$\int_{a}^{b} f(x) d x=\int_{\alpha}^{\beta} f(g(u)) g^{\prime}(u) d u$

[Any version of the fundamental theorem of calculus may be used providing it is quoted correctly.]

(b) Justifying carefully the steps in your argument, show that the improper Riemann integral

$\int_{0}^{e^{-1}} \frac{d x}{x\left(\log \frac{1}{x}\right)^{\theta}}$

converges for $\theta>1$, and evaluate it.

*Typos? Please submit corrections to this page on GitHub.*