Page 190 - ADVANCED MATH F.5

Form Five Form Five

1 1 1 1 1
5. − x cos2x + sin 2xc+ or x (sin )x 2 + cos sinx x − xc+
4 8 2 4 4
1 1
6. x 2 sin x + 2 cosx x − 2sin xc+ 7. 4 x 4 ln x − 16 x + 4 c
FOR ONLINE READING ONLY
8. 1 ( x − 2x ) ln 2x +− x 2 + c 9. ln 3x x −+
xc
2
x
2 4
1 2x − 3x  1 2 2 2 
10. e (2cos x + sin ) x + c 16. e −  x + x +  + c
5  3 9 27 
11. − 1 e − x (2cos 2x ( )) c+
( ) sin 2x+
5
2 1 x 1
12. e 2 (2sin x + cos ) x + c 17. x sin − 1 ( ) + 2x 1 4x + 2 c
−
5 2

13. 1 e 4x (2sin 2x − cos 2x ) c+ 18. 1 x 3 ln x − 1 x + 3 c
10 3 9

14. 1 e 2x (3sin 3x + 2cos3x ) c+ 19. 2 x ln x − 4 xc+
13
1 1
15. tanθ θ + ln cosθ + c 20. − ln x − + c
6x 6 36x 6
1 3 1 1 3 1 2
xc
21. (1 x+ ) ln 3x − ln x − x − x −+
3 3 9 2
n x
23. I = x e − nI n− 1 , I = 4 xe − 4 x 4x e + 3 x 12xe − 2 x 24xe + x 24e + x c
n

Integration using partial fractions
Teaching steps

1. Guide students to discuss the case where the integrand is a
proper rational function. Assist them to express a rational
function in terms of partial functions. Lead them to integrate
the partial fractions separately.
2. Assist students through discussion to consider the case where
an integrand is an improper rational function in which case the

184
Mathematics for Advanced Secondary Schools

30/06/2024 18:02:49
ADVANCED MATH F.5 TG CHAPTERS.indd 184
ADVANCED MATH F.5 TG CHAPTERS.indd 184 30/06/2024 18:02:49