Medium Practice — Set 1

Quick Recap — Techniques of Integration

• Substitution: $\int f(g(x))g'(x)\,dx=\int f(u)\,du$; e.g. $\int\tfrac{2x}{x^2+1}dx=\ln(x^2+1)$.
• By parts: $\int u\,dv=uv-\int v\,du$; ILATE order for choosing $u$.
• Standard: $\int\tan x\,dx=\ln|\sec x|$, $\int\cot x\,dx=\ln|\sin x|$.
• $\int\tfrac{dx}{x^2+a^2}=\tfrac1a\tan^{-1}\tfrac xa$, $\int\tfrac{dx}{\sqrt{a^2-x^2}}=\sin^{-1}\tfrac xa$.