Mathematics

Problem

(a) State the differentiation rule for the derivative of a composite function

F(g(x))

. (b) Consider the integral

\displaystyle\int 3x^{2}\,e^{x^{3}}\,dx

. Identify a suitable substitution

u=\,?

so that the entire integrand, including

dx

, can be rewritten wholly in terms of

u

. Write the transformed integral in terms of

u

(evaluation not required). (c) Give a brief explanation—using part (a)—of why replacing

u

x^{3}

in the antiderivative obtained in part (b) produces a valid antiderivative of the original integrand.

Intuitive explanation

The chain rule is the foundation connecting differentiation and integration: when you differentiate a composite function $F(g(x))$ , you multiply the outer derivative by the inner derivative, giving $F'(g(x)) \cdot g'(x)$ . For integration, $u$ -substitution reverses this process—you spot a function and its derivative hiding in the integrand, swap them for a single variable $u$ , and integrate the simpler form. Once you find the antiderivative in terms of $u$ , substituting back recovers the original variable because the chain rule guarantees the derivative will reconstruct your starting integrand perfectly.

Intuitive explanation

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