The idea

Integration by substitution

The formal version of the reverse chain rule — substitute u = g(x), replace dx using du = g'(x) dx, integrate in u, then convert back (or change the limits for a definite integral).

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Pure · Integration

Integration by substitution

The formal version of the reverse chain rule — substitute u = g(x), replace dx using du = g'(x) dx, integrate in u, then convert back (or change the limits for a definite integral).

Why it works

Rename the awkward inside

Integration by recognition works when you can see the answer — but sight fails on anything tangled. Substitution is the systematic version for when you can't: you rename the awkward inner part as a new variable

u

, and the whole integral simplifies into something standard. It is the reverse chain rule with the guesswork replaced by bookkeeping.

The mechanics rest on one substitution for the differential. If

u = g(x)

, then

\frac{du}{dx} = g'(x) \quad\Longrightarrow\quad du = g'(x)\,dx.

You treat

\dfrac{du}{dx}

like a fraction here (legitimate inside an integral): wherever

g'(x)\,dx

appears, replace it by

du

, and replace the inner

g(x)

u

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