0
Assessment
MAT1512 4
2026
, The A+Math Chicken
QUESTION 1
(a)
Evaluate:
∫(4𝑥 3 − 3𝑥 2 + 2) 𝑑𝑥
[3 marks]
ANSWER:
∴ ∫(4𝑥 3 − 3𝑥 2 + 2) 𝑑𝑥 = ∫ 4𝑥 3 𝑑𝑥 − ∫ 3𝑥 2 𝑑𝑥 + ∫ 2 𝑑𝑥
= 4 ∫ 𝑥 3 𝑑𝑥 − 3 ∫ 𝑥 2 𝑑𝑥 + ∫ 2 𝑑𝑥
𝑥 3+1 𝑥 2+1
= 4( )− 3( ) + 2𝑥 + C
3+1 2+1
4𝑥 4 3𝑥 3
= − + 2𝑥 + C
4 3
= 𝑥 4 − 𝑥 3 + 2𝑥 + C
(b)
Evaluate:
2
∫ 𝑥𝑒 𝑥 𝑑𝑥
[3 marks]
ANSWER:
Let 𝑢 = 𝑥 2
𝑑𝑢
= 2𝑥
𝑑𝑥
𝑑𝑢
𝑑𝑢 = 2𝑥 ∙ 𝑑𝑥 ⇒ 𝑑𝑥 =
2𝑥
, The A+Math Chicken
2 𝑑𝑢
∴ ∫ 𝑥𝑒 𝑥 𝑑𝑥 = ∫ 𝑥𝑒 𝑢 ( )
2𝑥
𝑒𝑢
= ∫ 𝑑𝑢
2
1
= ∫ 𝑒 𝑢 𝑑𝑢
2
1
= 𝑒𝑢 + C
2
1 2
= 𝑒𝑥 + C
2
(c)
Evaluate:
1
∫ 𝑑𝑥
𝑥In𝑥
[3 marks]
ANSWER:
Let 𝑢 = In𝑥
𝑑𝑢 1
=
𝑑𝑥 𝑥
𝑥𝑑𝑢 = 𝑑𝑥
1 1
∴∫ 𝑑𝑥 = ∫ (𝑥𝑑𝑢)
𝑥In𝑥 𝑥𝑢
1
= ∫ 𝑑𝑢
𝑢
= In|𝑢| + C
= In|In𝑥| + C
Assessment
MAT1512 4
2026
, The A+Math Chicken
QUESTION 1
(a)
Evaluate:
∫(4𝑥 3 − 3𝑥 2 + 2) 𝑑𝑥
[3 marks]
ANSWER:
∴ ∫(4𝑥 3 − 3𝑥 2 + 2) 𝑑𝑥 = ∫ 4𝑥 3 𝑑𝑥 − ∫ 3𝑥 2 𝑑𝑥 + ∫ 2 𝑑𝑥
= 4 ∫ 𝑥 3 𝑑𝑥 − 3 ∫ 𝑥 2 𝑑𝑥 + ∫ 2 𝑑𝑥
𝑥 3+1 𝑥 2+1
= 4( )− 3( ) + 2𝑥 + C
3+1 2+1
4𝑥 4 3𝑥 3
= − + 2𝑥 + C
4 3
= 𝑥 4 − 𝑥 3 + 2𝑥 + C
(b)
Evaluate:
2
∫ 𝑥𝑒 𝑥 𝑑𝑥
[3 marks]
ANSWER:
Let 𝑢 = 𝑥 2
𝑑𝑢
= 2𝑥
𝑑𝑥
𝑑𝑢
𝑑𝑢 = 2𝑥 ∙ 𝑑𝑥 ⇒ 𝑑𝑥 =
2𝑥
, The A+Math Chicken
2 𝑑𝑢
∴ ∫ 𝑥𝑒 𝑥 𝑑𝑥 = ∫ 𝑥𝑒 𝑢 ( )
2𝑥
𝑒𝑢
= ∫ 𝑑𝑢
2
1
= ∫ 𝑒 𝑢 𝑑𝑢
2
1
= 𝑒𝑢 + C
2
1 2
= 𝑒𝑥 + C
2
(c)
Evaluate:
1
∫ 𝑑𝑥
𝑥In𝑥
[3 marks]
ANSWER:
Let 𝑢 = In𝑥
𝑑𝑢 1
=
𝑑𝑥 𝑥
𝑥𝑑𝑢 = 𝑑𝑥
1 1
∴∫ 𝑑𝑥 = ∫ (𝑥𝑑𝑢)
𝑥In𝑥 𝑥𝑢
1
= ∫ 𝑑𝑢
𝑢
= In|𝑢| + C
= In|In𝑥| + C
