Page 199 - ADVANCED MATH F.5

Form Five Form Five

 1  1 cos x
θ 
22. tan 1  tan  + c 23. ln + c
 2  4 3 cos x − 3
θ
tan  + 1

FOR ONLINE READING ONLY
2
x 
24. ln  + c 25. 2 tan − 1   1 tan  + c

θ
tan  + 3 3  3  3

2

2sin x − 4 − 1
1) c
26. ln + c 27. tan (tan x ++
2sin x − 2
10 3   x
θ 
28. θ tan − 1  3 tan  + c 29. ln tan( ) 2+ 2 + c
−+
3   2
3
30. sin x 1 sin x+ − 1 3 (1 sin )x+ 2 + 1 sin xc+ +
Integration by splitting the numerator

Teaching steps
1. Guide students through discussion to recognize integrals
whose integrand is;
(a) A rational function with a quadratic denominator that
cannot be factorized.
(b) A rational function with trigonometric functions in both
the numerator and denominator.

2. Assist students through discussion to solve integrals of the
form:
±
⌠
(a)  ax b+ dx (b) ⌠ a cos xb sin x dx

⌡ cx + 2 dx e+ ⌡ p cos x q± sin x
a cos xb± ⌠ sin x c+ ⌠ a cos xb± sin x
(c)  dx (d)  dx
⌡ p cos x q± sin x ⌡ p cos x q± sin x r+
where a, b, c, d, e, p, q and r are constants.

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Mathematics for Advanced Secondary Schools

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