Senior 3 UEC Maths Name:

Integration
Note (1) 3. Integration by Partial Fraction
1 ∫ 𝑎 𝑑𝑥 = 𝑎𝑥 + 𝑐 Find
1
𝑎𝑥 𝑛+1 (a) ∫ 𝑥 2−9 𝑑𝑥
2 ∫ 𝑎𝑥 𝑛 𝑑𝑥 = + 𝑐, 𝑛 ≠ −1 𝑥
𝑛+1
1 (b) ∫ 𝑥 2−8𝑥+15 𝑑𝑥
3 ∫ 𝑥 𝑑𝑥 = ln|𝑥| + 𝑐
(𝑎𝑥+𝑏)𝑛+1
4 ∫(𝑎𝑥 + 𝑏)𝑛 𝑑𝑥 = +𝑐 Exercise (1)
𝑎(𝑛+1)
5 ∫ 𝑒 𝑥 𝑑𝑥 = 𝑒 𝑥 + 𝑐 Indefinite Integrals
𝑒 𝑓(𝑥) 1. Integrate the following.
6 ∫ 𝑒 𝑓(𝑥) 𝑑𝑥 = 𝑓′(𝑥) + 𝑐 (a) ∫ 12𝑥 −3 𝑑𝑥
𝑎𝑥 1
7 ∫ 𝑎 𝑥 𝑑𝑥 = ln 𝑎 + 𝑐 (b) ∫ 4√𝑥 − 3𝑥⁴ 𝑑𝑥
8 ∫ sin 𝑥 𝑑𝑥 = − cos 𝑥 + 𝑐 (c) ∫(4𝑥 − 1)2 𝑑𝑥
9 ∫ cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝑐 (d) ∫(1 − 𝑥)(3𝑥 2 + 𝑥 − 2) 𝑑𝑥
3
10 ∫ sec ² 𝑥 𝑑𝑥 = tan 𝑥 + 𝑐 (e) ∫ 𝑥 2 (2𝑥 2 + 3𝑥) 𝑑𝑥
11 ∫ cosec² 𝑥 𝑑𝑥 = − cot 𝑥 + 𝑐 𝑥 2 −5
(f) ∫ 𝑑𝑥
𝑥4
1
Example (1) (g) ∫ 3 √3𝑥 + 1 𝑑𝑥
Indefinite Integrals 2
(h) ∫( 3 cos 2𝑥 + 4 sec ²𝑥 − 5𝑒 𝑥 ) 𝑑𝑥
1. Find the following indefinite integrals. 1
(a) ∫ 4𝑥³ 𝑑𝑥 (i) ∫ (3−2𝑥)³ 𝑑𝑥
1 3
(b) ∫(𝑥 − 𝑥 )² 𝑑𝑥 (j) ∫ 5(2−3𝑥)² 𝑑𝑥
𝑥 4 +5 𝑥
(c) ∫ 𝑑𝑥 (k) ∫ sin 3 𝑑𝑥
𝑥
1
(d) ∫(2𝑥 2 + − 3 cos 𝑥) 𝑑𝑥 (l) ∫(3𝑒 𝑥 − 𝑒 −𝑥 ) 𝑑𝑥
𝑥
1 (m) ∫ 𝑒 2−3𝑥 𝑑𝑥
(e) ∫(3 𝑒 − 3 sin 𝑥 + √3 𝑎𝑥 ) 𝑑𝑥
𝑥
(n) ∫(𝑥 2 − 2𝑥)(𝑥 3 − 3𝑥 2 + 1)⁴ 𝑑𝑥
(f) ∫ tan2 𝑥 𝑑𝑥 𝑑𝑥
(o) ∫ 𝑥 ln 𝑥
(g) ∫ 4 sin 2𝑥 𝑑𝑥
𝑒 𝑥 𝑑𝑥
(h) ∫ 3√3𝑥 − 1 𝑑𝑥 (p) ∫ (1+𝑒 𝑥)³
4𝑥+3
(i) ∫ √2𝑥 2 𝑑𝑥
+3𝑥
15𝑥²
(j) ∫ 3 𝑑𝑥 2. Given 𝑓’’(𝑥) = 4𝑥 3 − 3, 𝑓(2) = −13 and
√5𝑥 +2
𝑓 ′ (−1) = −3, find 𝑓(𝑥).
(k) ∫ cot 𝑥 𝑑𝑥
1
(l) ∫ 𝑑𝑥 3. Find the following (Partial Fraction):
3𝑥−2
(m) ∫ 3𝑒 1−2𝑥 𝑑𝑥 1
(a) ∫ 𝑥(𝑥+2) 𝑑𝑥
𝑥
𝑑²𝑦 2
(b) ∫ (𝑥+2)(𝑥−7) 𝑑𝑥
2. Given = 9𝑥 − 1, 4𝑥−17
𝑑𝑥²
𝑑𝑦 (c) ∫ 2𝑥 2−3𝑥−5 𝑑𝑥
(a) Find , 7𝑥−1
𝑑𝑥
𝑑𝑦 (d) ∫ 1−𝑥² 𝑑𝑥
(b) Given 𝑑𝑥 = −7, and 𝑦 = 4 when 𝑥 =
𝑥 2 +3
−1, find 𝑦. (e) ∫ 𝑥 3−𝑥 𝑑𝑥
45𝑥+36
(f) ∫ 2 𝑑𝑥
35𝑥 +51𝑥+18
 5𝑥+22
(g) ∫ (𝑥−4)(𝑥+3)(𝑥+5) 𝑑𝑥 6.
3𝑥 2 +2𝑥−26
(a) Find the generated volume of region
(h) ∫ 𝑥 3−𝑥 2−10𝑥−8 𝑑𝑥 bounded by line 𝑦 = 𝑥², 𝑥 = 0, 𝑥 = 3 and
x-axis when it is revolved through 360°
4. Find about the x-axis.
(a) ∫ sin² 𝑥 𝑑𝑥
(b) ∫ 2 cos² 𝑥 𝑑𝑥
(c) ∫ cos² 4𝑥 𝑑𝑥
(d) ∫ cos² 𝑥 sin3 𝑥 𝑑𝑥
(e) ∫ sin2 𝑥 cos 2 𝑥 𝑑𝑥

Example (2)
Definite Integrals (b) In the diagram, the straight line KL is
𝑥²
𝑏
• ∫𝑎 𝑓(𝑥) 𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎) the normal to the curve 𝑦 = + 3 at
2
𝑃 (2,5). Find the volume generated, in
4. Evaluate: terms of 𝜋, when the shaded region is
3 3𝑥²
(a) ∫−1 𝑑𝑥 revolved through 360° about the x-axis.
5
−2 1
(b) ∫−3 𝑑𝑥
3𝑥+1
2
(c) ∫1 2𝑥√𝑥 2 − 1 𝑑𝑥
3
(d) ∫−5 𝑥 √4 − 𝑥 𝑑𝑥

Applications
𝑏 𝑑
• Area = ∫𝑎 𝑦 𝑑𝑥 / = ∫𝑐 𝑥 𝑑𝑦
𝑏 𝑑
• Volume = 𝜋 ∫𝑎 𝑦² 𝑑𝑥 / = 𝜋 ∫𝑐 𝑥² 𝑑𝑦 7. Given 𝑔’(𝑥) = 3𝑥, find the equation of
the curve 𝑔(𝑥) when it passes through
5. Find the area enclosed by (0,-6).
(a) the curve 𝑦 = 𝑥 2 − 1, the line 𝑥 = 1
and 𝑥 = 3.

(b) the curve 𝑦 = 𝑥 2 − 4, the y-axis and

the line 𝑦 = −3 and 𝑦 = 5.

(c) the curve 𝑦 = −𝑥² − 4𝑥 and the x-axis.

Miscellaneous Exercise (i) the equation of the curve.
1 Evaluate the following: (ii) the value of k.
𝜋
a) ∫012 sin 3𝑥 𝑑𝑥 (b) A curve has a turning point (2, 1) and
𝜋 𝑝
b) ∫0 cos ⁵𝑥 𝑑𝑥 its gradient function is given by − 𝑥² ,
𝜋
c) ∫0 sin ²𝑥 cos 𝑥 𝑑𝑥 where 𝑝 is a constant. Find
𝜋 (i) the value of 𝑝.
d) ∫𝜋2 (sin 𝑥 + cos 𝑥)² 𝑑𝑥 (ii) the equation of the curve.
3
𝜋
e) ∫0 sin3 𝑥 𝑑𝑥
3 (c) The equation of the tangent to a curve
at point (1, k) is 𝑦 = 5𝑥 − 7. The gradient
function of the curve is 6𝑥² − 𝑛𝑥, where 𝑚
2 Evaluate:
3 2𝑥+1 is a constant. Find
a) ∫−1 𝑥 𝑑𝑥 (i) the values of 𝑘 and 𝑛.
2
b) ∫0 (𝑥 2 + 1)(𝑥 − 3)𝑑𝑥 (ii) the equation of the curve.
6 1
c) ∫2 𝑑𝑥
√2𝑥−3
8 𝑥²
6 Answer the following questions.
d) ∫−1 √𝑥+1 𝑑𝑥 (a)
6 1
e) ∫2 2𝑥−3 𝑑𝑥

dy
3 (a) If 𝑦 = 𝑥 √6 + 3𝑥², prove that =
dx
6+6𝑥²
. Hence or otherwise, find the value
√6+3𝑥²
5 1+𝑥²
of ∫1 𝑑𝑥.
√6+3𝑥²
dy
(b) If 𝑦 = (2𝑥 − 1)√𝑥 − 2, prove that =
dx
3(2𝑥−3)
. Hence or otherwise, find the value
2√𝑥−2
6 2𝑥−3
of ∫3 𝑑𝑥.
√𝑥−2
(b)
3
4 (a) If ∫1 𝑓(𝑥) 𝑑𝑥 = 5, find
1
(i) ∫3 𝑓(𝑥) 𝑑𝑥
1
(ii) ∫3 [𝑓(𝑥) + 2] 𝑑𝑥
2 3
(iii) ∫1 3𝑓(𝑥) 𝑑𝑥 + ∫2 3𝑓(𝑥) 𝑑𝑥
0
(b) If ∫𝑎 𝑥(2 − 3𝑥) 𝑑𝑥 = −2, find the value
of 𝑎.
𝑑 𝑥³
(c) Given that ( ) = 𝑓(𝑥), find the
𝑑𝑥 𝑥 2−1
2 10
value of 𝑚 if ∫0 [𝑓(𝑥) − 𝑚𝑥] 𝑑𝑥 = − . (c)
3

5 (a) The gradient function of a curve is

given by 𝑓′(𝑥) = (1 − 2𝑥)³. The curve
1
passes through the points (1,− ) and (2,
2
k). Find
(d)