Consider the composite function

MATHEMATICS · Subjective · A Level · Mathematics · Integration · Integration

Problem

H(x)=f(g(x))

, where

f

and

g

are twice-differentiable on an open interval containing

x=3

and

g'(x)\ne0

on that interval. (a) Write a clear, logical argument that establishes a relationship between

\displaystyle\int f'(g(x))\,g'(x)\,dx

and an antiderivative of

H(x)

. (b) Explain why the differential expression

g'(x)\,dx

can be interpreted as the differential of

g(x)

even though

dx

is not itself a number. (c) Describe, without performing any integrations, the conditions under which replacing

g'(x)\,dx

by a single differential variable would fail to produce a valid antiderivative rule.