An Integral Built from an Inverse
Difficulty: Hard · Topics: Calculus, Inverse Functions, Integration
Teskari funksiyadan tuzilgan integral
g(x) is the inverse of f(x). Both are differentiable functions on mathbbR and satisfy the following. (a∈mathbbR)
f(1)=8, ∫ 1^a(1)/(g'(f(x)) f(x)) dx=2ln a+ln(a+1)-ln 2.
Find f(2).
g(x) — f(x) ning teskarisi. Ikkalasi ham mathbbR da differensiallanuvchi va quyidagilarni qanoatlantiradi. (a∈mathbbR)
f(1)=8, ∫ 1^a(1)/(g'(f(x)) f(x)) dx=2ln a+ln(a+1)-ln 2.
f(2) ni toping.