Paper 1, Section I, E

Analysis I | Part IA, 2020

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

abf(x)dx=αβf(g(u))g(u)du\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

0e1dxx(log1x)θ\int_{0}^{e^{-1}} \frac{d x}{x\left(\log \frac{1}{x}\right)^{\theta}}

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

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