Integrating Using Substitution AB & BC

Substitution reverses the chain rule. Set $u$ to the inside of a composition whose derivative appears as a factor, compute $du$, and rewrite so that no $x$ survives; if one does, the substitution has not fitted and continuing produces an expression in two variables. A missing constant factor can be supplied, since constants pass through an integral, so $\int x(x^{2}+1)^{4}dx = \frac{1}{2}\int u^{4}du$. A missing variable factor cannot: moving a $\frac{1}{2x}$ across the integral sign is not a legal step, and $\int (x^{2}+1)^{4}dx$ is simply a different problem.

With limits attached there are exactly two correct finishes. Convert each $x$-limit through $u$ and evaluate entirely in $u$, or return to $x$ first and use the original limits. Doing both, or neither, is the error, and doing neither is the silent one: evaluating $\left[\frac{u^{5}}{5}\right]$ at $x = 0$ and $x = 2$ instead of at $u = 1$ and $u = 5$ gives $6.4$ rather than $624.8$, an ordinary-looking number with nothing visibly wrong. Converted limits need not increase, and swapping them to make them do so negates the result.