AHL 5.15 (HL)—Further derivatives and integrals

Further derivative pairs unlock advanced integrals.

Key derivatives are $(\tan x)'=\sec^2x$, $(\sec x)'=\sec x\tan x$, $(\cosec x)'=-\cosec x\cot x$, $(\cot x)'=-\cosec^2x$, $(a^x)'=a^x\ln a$, $(\log_a x)'=1/(x\ln a)$, $(\arcsin x)'=1/\sqrt{1-x^2}$, $(\arccos x)'=-1/\sqrt{1-x^2}$ and $(\arctan x)'=1/(1+x^2)$.

Example

Because $1/[(x+1)(x+2)]=1/(x+1)-1/(x+2)$, partial fractions give $\int dx/[(x+1)(x+2)]=\ln|x+1|-\ln|x+2|+C$. For $\int dx/[1+(2x+1)^2]$, the linear inner derivative gives $\tfrac12\arctan(2x+1)+C$.

Match an integrand to a derivative pair, include the reciprocal inner-gradient factor for a linear composite, and use partial fractions before integrating a rational expression when required.

Inverse-trig derivative domains and logarithmic absolute values matter; a memorised form without its domain or inner-gradient factor is incomplete.