RVHS H2 Math Integration Techniques Qns

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Integration Techniques 1 Integration Techniques 1. AJC/2009/I/8 Find (i) ( ) 2 1 2e tan e dxx x−−  , [3] (ii) 2 d 1 4 2 x x xx−− . [3] 2. CJC/I/3 Find (i) xxx d sin , [2] (ii) x xx d 1 1  − , where 10  x , using the substitution ux 2cos= . [4] 3. PJC/I/10 (a) Use the fact that 5sin𝑥 − 3cos𝑥 = (cos𝑥 + sin𝑥) − 4(cos𝑥 − sin𝑥) to find the exact value of ∫ 5sin𝑥−3cos𝑥 cos𝑥−sin𝑥 𝑑𝑥 0 𝜋 6 . [4] (b) Find ( ) 2d 1d xxx − , expressing your result in the form 2 21 a bx x + − , where a and b are constants to be determined. [2] Hence, or otherwise, obtain the exact value of 1 2 2 2 0 32 d 1 x x x − − . [3] 4. MJC/I/3 Using the substitution 2etx = , show that ln ln 2 2 d 2 d ln ln 2 2 2 x x t t x x t − = − +− − −  . Hence evaluate 4 2 2e 2e ln ln 2 d ln ln 2 2 x x xx − −− , giving your answer in the form b a 2 where 𝑎, 𝑏 ∈ ℤ. [7]

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Integration Techniques 2 5. MJC/II/1 (i) Differentiate cose x with respect to x. [1] (ii) Find cose sin 2 dx xx . [3] 6. SAJC/I/8 (a) Show that 2 2 2 5 13 25 xx xx −+ −+ can be written in the form 2 (2 2) 25 B x CA xx −++ −+ where A, B and C are constants to be determined. Hence find the integral ∫ 2𝑥2−5𝑥+13 𝑥2−2𝑥+5 d𝑥. [5] (b) Find, in terms of n and e, ∫ 𝑥𝑛−1 ln 𝑥 d𝑥  2e  1 . [4] 7. ACJC/2010/II/1 Find the exact value of ∫ |e2𝑥 − 1 e2(𝑥−1)| 1 −1 d𝑥. [4] 8. AJC/2011/I/1 Find 12cos dx x x −  . [3] 9. IJC/2011/I/3 Using the substitution , find . 10. NYJC/2011/I/2 (a) Sketch the graph of  y = x − a 2 for  −a  x  a and  a  0 . [1] Hence, find k such that 0 0 d d22 a a aax x k x x − − = − . [4] (b) Find (i) 2d ( sin 2 )d xxx , (ii) 2 cos 2 d x x x . [3] 1 e2 ux = ( ) ( )  2 2 ln 2 d 25 2 ln 2 x x xx  − 

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Integration Techniques 3 11. ACJC/2011/I/3 (a) Find (i) ( ) 2tan dx x x , [2] (ii) 2 d3 x xxx ++ . [4] (b) Evaluate, exactly, (i) ( ) 1 122 0 sin dx x x−  , [4] (ii) 1 0 dx x b x− where 01 b . [3] 12. DHS/2012/I/2 Obtain a formula for ( ) 1 2 21 2 tan 2 d14 n x xx − + in terms of n, where 0.n  Hence evaluate ( ) 1 2 21 2 tan 2 d14 x xx − + exactly. [4] 13. MJC/2012/I/3(b) (i) Find ( ) 2d 2d x x . [1] (ii) Hence find 22 ln 2 dxx x . [3] 14. SRJC/2012/I/12 (a) Find the integral 2 (ln ) d .2 x x [3] (b) Solve the inequality x ax 4 3  , leaving your answers in terms of a, where 13 a . Hence find 43 3 1 d , axx x− in terms of a. [6] 15. CJC/2013/I/2 Find the value of k such that 02 0 dd c c x c x k x c x − − = − , where c is a positive constant. 16. DHS/2013/II/1

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Integration Techniques 4 (a) Find ( )( ) 2 d12 x xxx−− . [3] (b) (i) Differentiate ( ) 12sin x− with respect to x. [1] (ii) Hence or otherwise, find a positive integral value of n such that ( ) 12 0 1sin d 42 n x x x − =− . [3] 17. DHS/2014/I/8 (a) Find 2sec ( ) d ,+ x x a x where a is a constant. [3] (b) Find 2 1 d22 − −+ x xxx . [2] Hence find (i) the exact value of 2 21 4 d,22 − −+ x xxx [4] (ii) 22 1 d22 p p x xxx− − −+ where p is a constant, 1.p Leave your answer in terms of p. [3] 18. HCI/2014/I/2 By using the substitution 1ux=− , show that ( ) ( ) 11 00 1 d 1 d mnnmx x x x x x− = − . [2] Hence, or otherwise, evaluate 1 2 0 1dx x x − , express your answer in exact form. [3] 19. JJC/2014/I/6 (a) (i) Obtain a formula for 21 1 ln d n xx x in terms of n, where n > 1. [3] (ii) Hence evaluate 21 1 ln dxx x   . [1] [You may assume that 1 ln 0 as nnn → →  .] (b) Use the substitution secxa = to find the exact value of 2 22 a a xa dxx −  in terms of a and , where a is a positive constant . [4] 20. PJC/2014/I/7 It is given that

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Integration Techniques 5 2 2 1 0 , 1+f ( ) 2 5, xa xx ax  =   −   and that ( ) ( )f 5 fxx+= for all real values of x , where 05 a . (i) Show that ( )f 100 1 = . [1] (ii) Sketch the graph of f ( )yx= for 2255 a x a− +   + . [3] (iii) By using the substitution 1ux=+ , find 2 0 f ( ) d a xx , in terms of a. [5] 21. ACJC Prelim/2020/01/Q1 (i) Use the substitution u = 2x – 1 to find 2 d 1 (2 1) x x x−− . [4] (ii) Hence find 1sin (2 1) dxx− − . [2] 22. DHS/2022/I/Q3 (a) Differentiate 2sin 2e x with respect to x. [2] (b) Find 2 2sin 2 sin 2e sin 4 d. 1+e x x x x   [2] (c) Find the exact value of [3] 23. DHS Prelim 9758/2024/01/Q4 (a) Find 2 14 8 3 d 9 x x x x−− +  . [4] (b) Find 2 2 0 3 e dkxxx in terms of k, where k is a positive constant. Explain whether there exist solutions for k satisfying the equation 2 2 30 63 e d .kxxx k=− [4] 24. VJC Prelim 9758/2024/01/Q6 4 2 0 2sin 2 e sin 4 cos 2 d . x x x x  

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Integration Techniques 6 (a) Find the exact value of 3 12 21 2 2sin d 1 x x x − −   . [3] (b) Find the exact value of π 3 0 cos2 dxx . [3] (c) Find 22 1 d23 xx kx k− + +  , where k is a positive constant. [4] 25. ASRJC Prelim 9758/2025/P2/Q5 (a) Solve the following integral. (i) 2 12 d 32 x x xx − +−   . [2] (ii) ( ) 2ln 2 dx x x − , where 22 x−   . [3] (b) A function f is defined by f ( ) e 2 xx =− . (i) Sketch the graph of f ( )yx= , indicating clearly the equation(s) of asymptote(s), if any. [2] Hence find ln 3 0 f ( ) dxx , giving your answer in exact form. [3] 26. NYJC Prelim 9758/2025/P1/Q7 (a) Use the substitution 2sinx = , where 0 2  , to find 1 2 0 16 d1 x xx− exactly. [4] (b) Find 2 1 d 1 12 b a x x xx − −+  in terms of a and b, where 1ab . [4] Answers 1 (i) ( ) ( ) 2 1 2 411 tan ln 124 x x xe e e C−− + + + (ii) ( )21 2111 1 4 2 sin2 23 xx x C − +− − − − + 2 (i) sin cosx x x c−+ (ii) - 112ln x cxx  −++ 3 (a) 2 3 1 ln32  −+  ; (b) 2 2 12 1 x x − − , 3 34  + 4 16 2 3 , 16=a , 3=b

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Integration Techniques 7 5 (i) cose sinx x− (ii) cos cos2e cos 2exx xC− + + 6 (a) 211 1 12 ln 2 5 2 tan2 2 2 xx x x c − −   − − + + +        (b) ( ) ( ) ( )2 1 2 ln 2 1 2 1 nn n e en  + − + 7 ( ) 4 2 21 412 e e e e −+ − + + 8 2 1 2 4 1cos 122 x x x C− − − + 9 ( ) ( ) ( )5 2 ln 215 ln ln 22 2 2 5 2 ln 2 x xc x  + −+ − 10 (b) (i) 22 sin 2 2 cos 2x x x x + (ii) 21 1 1cos 2 sin 2 sin 22 4 2x x x x x C− + + 11 (a)(i) ( ) 21 ln cos2 xc−+ (a)(ii) ( )21 2111ln 3 tan2 11 11 xx x c − ++ + − + (b)(i) 31 24 4 2  +− (b)(ii) 3 1 3 3 2 bb +− 12 ( ) 3311 tan 264 n − −  , 37 384  13 (i) 212 ln 2x+ , (ii) 21 12 2ln 2 x xC−  −+ 14 (a) 2(ln )