answer tutorial MAT235 Chapter 1 (Integration Techniques)

MAT235- CALCULUS II FOR ENGINEERS/ Chapter 1 : Integration Techniques

Tutorial 1 : Integration Techniques

1. MAT235/ Dec 2019/ Q1

a) Solve ∫ 2 −2 by using integration by parts. (7 marks)

Ans : − 1 2 −2 − 1 −2 – 1 −2 +

2 24

b) Use the trigonometric substitution = 3 to evaluate ∫ 2 . (9 marks)
√36+4 2

Ans : √9 + 2 + C

c) Solve ∫ 2 +8 by using partial fractions. (7 marks)
( +3)( −1)

Ans : −1 | + 3| + 5 | − 1| +

2 2

2. MAT235/ Jun 2019/ Q1

∶ −1 − 1 |1 + 2| +
2

∶ √ 2 − 16 − 4 −1 +
(4)

∶ = 1 , = 2 , = −1

∶ | | − 2 − | + 1| +


© Amirah 2022 110

MAT235- CALCULUS II FOR ENGINEERS/ Chapter 1 : Integration Techniques

3. MAT235/ Dec 2018/ Q1

∶ 1 ( 2 + 1) −3 − 2 −3 − 2 −3 +
−3 9 27

∶ 1 (√9 − 5 − 1 (√9 − 3 +
5 3
2) 2)

∶ 1 |2 + 1| − 1 1 − 1 1)2 +
2 2 + 2(2 +

4. MAT235/ Jun 2018/ Q1

∶ −1(2 ) + 1 √1 − 4 2 +
2

∶ 1 (√1 + 4 3 +
12
2)

∶ = −1 , = 1 , = 2

∶ − | | + 1 | 2 + 1| + 2 −1( ) +
2

© Amirah 2022 111

MAT235- CALCULUS II FOR ENGINEERS/ Chapter 1 : Integration Techniques

5. MAT235/ Jan 2018/ Q1

∶ | + 2| + 2 | + 2| − − 2 +

∶ 18.33

11 3
∶ 3 − 3 + 3 − (3 + 3)2

∶ 1 | | − 1 | + 1| + 1 1) +
3 3 3( +

6. MAT235/ Mar 2017/ Q1

∶ − 1 ( 2 + 1) 3 + 2 + 2
3 9 3 27 3 +

∶ 1 (√4 + 5 − 4 (√4 + 3 +
5 3
2) 2)

∶ = 0 , 1 11
= 2 , = 8 , = − 8

11 1
∶ − 2 + 8 | − 2| − 8 | + 2| +

© Amirah 2022 112

MAT235- CALCULUS II FOR ENGINEERS/ Chapter 1 : Integration Techniques

7. MAT235/ Oct 2016/ Q1

∶ −1(2 ) − 1 |1 + 4 2| +
4

∶ 4 −1 + 1 √16 − 2 +
(4) 4

∶ 1 5(2 ) − 1 3 (2 ) +
10 6

Q2
.

∶ = 1 , = 1 , = −3

∶ | | + 1 |1 + 2| − 3 −1( ) +
2

8. MAT235/ Mar 2016/ Q1

∶ −1 − 2 |16 + 2| +
(4)

∶ 3√ 2 − 25 +

∶ 1 7 − 1 5 +
7 5

© Amirah 2022 113

MAT235- CALCULUS II FOR ENGINEERS/ Chapter 1 : Integration Techniques

Q2.

31 7
∶ ) + + 1 − ( + 1)2

∶ ) 3 | | + | + 1| + 7
+1+

9. MAT235/ Sep 2015/ Q1

∶ 9 −2 [3 (3 + 1) − 2 (3 + 1)] +

52

11
∶ 8 − 32 4 +

∶ √ 2 − 25 +

10. MAT235/ Mar 2015/ Q1

∶ 2
[2 (2) − (2)] +
5

∶ 1 4( ) + 1 6 ( ) +
4 6

∶ 2√ 2 − 36 +

© Amirah 2022 114