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2022

MAT2691-MATHEMATICS II [ENGINEERING]




SEMESTER 01
ASSIGNMENT 02 SOLUTIONS
Operational Research
P.hD..Msc..Hons..Bsc in
Mathemetics
Zakes
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, Question 1

1.1
𝜋
∫ 𝑥 3 √ln 𝑥 𝑑𝑥

Let 𝑢 = √ln 𝑥
1
𝑑𝑢 = 𝑑𝑥
2𝑥√ln 𝑥

𝑑𝑥 = 2𝑥√ln 𝑥 𝑑𝑢
𝜋
= ∫ 𝑥 3 𝑢 2𝑥𝑢 𝑑𝑢
2𝜋
=∫ 𝑑𝑢
𝑥2
2
If 𝑢 = √ln 𝑥 then 𝑥 = 𝑒 𝑢
2𝜋
=∫ 2 𝑑𝑢
(𝑒 𝑢 ) 2

2𝜋
=∫ 2 𝑑𝑢
𝑒 2𝑢
1
= 2𝜋 ∫ 2 𝑑𝑢
𝑒 2𝑢

Integrate by parts
1 2
𝑙𝑒𝑡 𝑢 = 2 𝑢′ = −4𝑒 −2𝑢 𝑢 𝑣′ = 1 𝑣=𝑢
𝑒 2𝑢

𝑢 2
= 2𝜋 ( 2 − ∫ −4𝑒 −2𝑢 𝑢2 𝑑𝑢)
𝑒 2𝑢

𝑢 2 √𝜋
= 2𝜋 ( 2 − (𝑒 −2𝑢 𝑢 − (√2𝑢)))
𝑒 2𝑢 2√2


√ln 𝑥 2 √𝜋
= 2𝜋 ( 2 − (𝑒 −2(ln 𝑥) √ln 𝑥 − (√2 ln 𝑥)))
𝑒 2 (√ln 𝑥) 2 √2


𝜋√𝜋
= (√2√ln 𝑥) + 𝐶
√2




1.2
𝑒
sin−1 𝑥
∫04 √1−𝑥2 𝑑𝑥

Let 𝑢 = sin−1 𝑥
𝑑𝑢 1
=
𝑑𝑥 √1−𝑥 2

𝑑𝑥 = √1 − 𝑥 2 𝑑𝑢
𝑢
∴∫ . √1 − 𝑥 2 𝑑𝑢
√1−𝑥 2




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