Recognising Standard Integrals

Integration reverses differentiation. For an indefinite integral, include a constant \(C\) because every antiderivative differs by a constant.

\(\int\frac{f'(x)}{f(x)}\,\mathrm dx=\ln|f(x)|+C\) on intervals where \(f(x)\ne0\).

\(\int\frac{1}{1+x^2}\,\mathrm dx=\arctan x+C\), and \(\int\frac1{\sqrt{1-x^2}}\,\mathrm dx=\arcsin x+C\) for \(|x|<1\).

For a linear input \(ax+b\), divide by \(a\) after applying the standard integral. Differentiate the proposed answer to check the missing factor.