DrFrostMaths
C4 Integration Cheat Sheetf(x)How to deal with itfxdx (+constant)Formula booklet?sin xStandard result-cosxNocos xStandard resultsinxNotan xIn formula booklet, but use sinxcosxdx which is of the form kf'xfxdxlnsecxYessin2 xFor both sin2x and cos2x use identities for cos2xcos2x=1-2sin2xsin2x=12-12cos2x12x-14sin2xNocos2 xcos2x=2cos2x-1cos2x=12+12cos2x12x+14sin2xNotan2x1+tan2x≡sec2xtan2x≡sec2x-1tanx-xNocosec xWould use substitution u=cosec x+cotx, but too hard for exam.-lncosec x+cotxYessec xWould use substitution u=secx+tanx, but too hard for exam.lnsecx+tanxYescot xcosxsinxdx which is of the form f'xfxdxlnsinxYescosec2xBy observation.-cotxNo!sec2xBy observation.tanxYes (but memorise)cot2x1+cot2x≡cosec2x-cotx-xNosin 2x cos 2xFor any product of sin and cos with same coefficient of x, use double angle.sin2xcos2x≡12sin4x-18cos4xNo1xlnxNolnxUse IBP, where u=lnx,dvdx=lnxxlnx-xNoxx+1Use algebraic division.xx+1≡1-1x+1x-lnx+11xx+1Use partial fractions.lnx-lnx+14xx2+1Reverse chain rule. Of form kf'xfx2lnx2+1xx2+12Power around denominator so NOT of form kf'xfx . Rewrite as product.xx2+1-2Reverse chain rule (i.e. “Consider y=x2+1-1" and differentiate)-12x2+1-1e2x+111-3xFor any function where ‘inner function’ is linear expression, divide by coefficient of x12e2x+1-13ln1-3xx2x+1Use sensible substitution. u=2x+1 or even better, u2=2x+1.1152x+1323x-1sin5x cos xReverse chain rule.16sin6x ................
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