DHS Integration & Its Applications (9758) Topical Revision

7. Integration and its Applications 1 7. Integration and its Applications 1 CJC/2010Promo/10 Find (a) d3 x xx [2] (b) 2 ln dx x x [3] (c) 2 3 d4 7 x xx x    [4] (d) 2 2 1 d 4 x x x    by using the substitution x = 2 sin . [4] 2 DHS/2009Promo/9 (i) Find d (sin 2 ).d xx [1] (ii) Find   2 sin cos d . cos sin x x x x x   [2] (iii) Hence find the exact value of   6 2 0 sin sin 2 cos sin 2 d cos sin x x x x x x x    . [4] 3 ACJC/2010Promo/9 Find the following integrals. (a) 2cos3 cos 3 sin 3 cot 3 x ec x dxx x   [2] (b) 2 1 1 16 x dx x    [4] (c)   2 1 lnx x dx   [4] 4 TJC/2009Promo/2 Integrate the following: (a) 1 x ln x2 dx , where x > 0. [2] (b) 2 1e d 2 1 x x x   , where x > 1 2 . [2]

7. Integration and its Applications 2 5 Using the property    4 2 2sec sec ,sec   find 4sec d  . [3] 6 JJC/2010Promo/12 (a) Write down constants A and B such that, for all values of x , 4 (2 6)x A x B    . [2] Hence find 2 4 d 6 13 x x x x    . [4] (b) Using the substitution 1x u , show that 1 1 4 2 3 1 2 4 1 e d d ex ux u x u  . [2] Hence evaluate 1 4 3 2 1 e dx x x , giving your answer in exact form. [3] (c) Find the exact value of 0 3 3 2 dx x   . [3] 7 RVHS/2010Promo/11 (modified) (a) Using the substitution exu , find 1 de 2ex x x . [4] (b) Show that 1 20 4 5 3 π d = 63 2 x a b x x x      , where a and b are constants to be found. [6] 8 VJC/2013Promo/5 (a) (i) Prove that   22 2 2 d 2 1 d 1 1 1 x x x x x          . [2] (ii) Find the exact value of   1 22 0 1 d . 1 x x  [3] (b) Find the constant A such that Hence find [3] 2 2 2 1 e .1 e 1 e x x x A   2 1 d .1 e x x

7. Integration and its Applications 3 9 HCI/2020Prelim/I/6 Find (a) 3e d5 0.3e x x x , [2] (b)  cos ln dx x , where 0x , [3] (c) the exact value of 3 0 2e 5 dx x .