ACJC 2026 Integration Techniques Summary

Keep a , write it as and use Use double angle formulae to rewrite integrand: OR Keep sin x or cos x as and use Powers of sin x & cos x Special case Powers of tan x Integration by parts MF27 START Denominator CANNOT be factorised 2026 H2 Mathematics 9758: Integration Techniques / Summary / Page 1 of 2 Alternative method Write numerator as “ (derivative of denominator)km + ” Find k & m. Then use f ( ) d ln f ( )f ( ) x x x Cx  =+ and 1 22 11 d tan() xBxC AAA x B − −=+ +−   Identify the differentiable function (u) using L – Logarithms I – Inverse Trigonometry A – Algebraic e.g. powers of x T – Trigonometric function E – Exponential function & the integrable function d d v x  Then dd dddd vuu x uv v xxx =− Basic Trigo For (exp)(trigo) dx , either choice works. Do by parts TWICE & make a subject of integral. Anglo-Chinese Junior College Resolve into partial fractions and integrate Write numerator as “ (derivative of denominator)km + ” Find k & m. Then use     1 f ( )f ( ) f ( ) d 1 n n xx x x C n +  =+ + 1 22 1 d sin () xBxC AA x B − −=+ −−   linear quadratic INTEGRATION TECHNIQUES constant quadratic or constant quadratic Int. into Inv. Trigo F product of factors d dd vux x  Denominator CAN be factorised of the form B MF27 1 22 1 22 11 d tan 1 d sin xxC aaxa xxC aax − − • = ++ • = + −   C MF27 22 22 11 d ln 2 11 d ln 2 axxC a a xax xaxC a x axa +• = + −− −• = + +−   2 2 1 d 1 d x ax bx c x ax bx c • ++ • ++   Alternative If 2ax bx c++ CAN be factorised E G A Int. into ln 22 22 1 d 1 d xax x ax •  • −   D Complete the square and 2ax bx c++ 22 22 1 d () 1 d () x A x B x A x B • − • −−   

Use double angle formulae Keep sin x as f ′(x) Basic Trigo Powers of sin x & cos x Special case Denominator CANNOT be factorised INTEGRATION EXAMPLES or Inv. Trigo Powers of tan x F product of factors Denominator CAN be factorised of the form MF27 Integration by parts E G A B C D START MF27 MF27 ln 2026 H2 Mathematics 9758: Integration Techniques / Summary / Page 2 of 2 Alternative Resolve into partial fractions and integrate Alternative method e sin d e sin e cos d e sin e cos e sin d e sin e cos e sin d 2 e sin d e sin e cos 1 e sin d e (sin cos )2 x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x C x x x x K =− = − − −  = − −  = − + = − +      22 22 22 11 ln d ( )(ln ) d 1 (ln ) d22 ln 24 e d (e )( ) d sin d (1)(sin ) d ln d (1)(ln ) d xx x x x x x x xx xx x xx xC x x x x x x x x x x x x • • −−• • = =−  = − + = = =      Anglo-Chinese Junior College Complete the square 2 2 1 d 2 15 1 d ( 1) 16 x xx x x • +− = +−   2 22 1 d 82 1 d 3 ( 1) x xx x x • −− = −+     f ( ) f ( ) , 1 n x x n − ( ) 22 22 21 31 22 31 24 (2 1)2 dd11 (2 1)1 3 1 dd22 1 1 2 1ln 1 3 tan2 3 xx xxx x x x x xxxx x xx x C − +++ =+ + + + +=+ ++ ++ += + + + +        22 22 21 1 2 1 2 ( 2 2) 1dd 8 2 8 2 ( 2 2) 1 d d 8 2 9 ( 1) 18 2 sin 3 xx xx x x x x x xx x x x xx x C − − − − −= − − − − − − −=− − − − + +=− − − − +        