Uvero — STEM Learning

Integration by Substitution

Substitution reverses the chain rule by replacing a repeated inner expression and its differential. Choose $u=g(x)$ so that $g'(x)dx$ appears up to a constant factor. The distinction between a formal hypothesis and an intuitive picture is made explicit so that calculations can be justified, not merely patterned.

Worked reasoning

Let $u=x^2$ , so $du=2x\,dx$ .
$\int\cos u\,du=\sin u+C=\sin(x^2)+C$ .

Evaluate $\int 2x\cos(x^2)\,dx$ .

Which statement best captures the central mathematical idea in Integration by Substitution?

Choose $u=g(x)$ so that $g'(x)dx$ appears up to a constant factor.

A familiar formula remains valid after its hypotheses, domain or convergence conditions are removed.

A memorised formula is valid even when its conditions are not met.

The diagram or notation replaces the need to justify each mathematical step.

When starting a problem about Integration by Substitution, which move is most reliable?

Write the governing definition and list its hypotheses before calculating in integration by substitution.

Round every value before identifying the mathematical structure.

Choose the longest available formula and substitute immediately.

Assume the desired conclusion and work backwards without checking equivalence.

Which statement is a misconception that must be rejected when working with Integration by Substitution?

A familiar formula remains valid after its hypotheses, domain or convergence conditions are removed.

Choose $u=g(x)$ so that $g'(x)dx$ appears up to a constant factor.

Write the governing definition and list its hypotheses before calculating in integration by substitution.

Test a representative example and a boundary or counterexample, then verify the conclusion directly from the definition.