Substitution and Definite Bounds

Choose \(u=g(x)\) when an integrand contains both a composite expression and a multiple of its derivative. Replace the differential as well as the expression.

If \(u=g(x)\), then \(\mathrm du=g'(x)\,\mathrm dx\). For a definite integral, replace each \(x\)-bound with its corresponding \(u\)-value before evaluating.

A prescribed AA HL substitution may be supplied in an examination when it is not a direct \(g'(x)f(g(x))\) pattern. Finish in the original variable for an indefinite integral.

Check an indefinite result by differentiating; check a definite result against the sign and approximate size of the area when the integrand has a fixed sign.