Selecting Techniques for Antidifferentiation AB & BC

With every technique available, the work is choosing one, and four questions in order do it: is it a basic form, would algebra or an identity simplify it, is some inside function's derivative present so that substitution fits, and is it a product of unlike pieces calling for parts. The order matters, since $\int \frac{x^{2}+1}{x}dx$ is a one-line split and a wasted substitution otherwise. A fifth outcome is legitimate: $\int e^{x^{2}}dx$ and $\int \frac{\sin x}{x}dx$ have no elementary antiderivative, and recognising that is an answer rather than a defeat.

Near-identical integrands separate by one symbol. $\int x e^{x^{2}}dx$ is a substitution and $\int x e^{x}dx$ is parts, because in the first the $x$ is the derivative of an inside and in the second there is no inside. $\int \frac{x}{x^{2}+1}dx$ gives a logarithm, $\int \frac{dx}{x^{2}+1}$ an arctangent, and $\int \frac{x^{2}}{x^{2}+1}dx$ needs dividing first. When an attempt fails, read what it condemns: a stray variable condemns the choice of $u$ and not the method, a harder integral from parts condemns the split of $u$ and $dv$, and the original integral reappearing is not a failure at all.