Mathematics III (Engineering)
Assignment 01 Answers
Semester 1 2025
0027 63 985 5033
,
,QUESTION 1.
𝑑𝑦
cos 𝑦 = 2 𝑠𝑖𝑛2 𝑥 cos 𝑥
𝑑𝑥
∫ cos 𝑦 𝑑𝑦 = 2 ∫ 𝑠𝑖𝑛2 𝑥 cos 𝑥 𝑑𝑥
sin 𝑦 = 2 ∫ 𝑠𝑖𝑛2 𝑥 cos 𝑥 𝑑𝑥
For the integral
∫ 𝑠𝑖𝑛2 𝑥 cos 𝑥 𝑑𝑥 𝑙𝑒𝑡 𝑢 = sin 𝑥 𝑑𝑢 = cos 𝑥
∫ 𝑠𝑖𝑛2 𝑥 cos 𝑥 𝑑𝑥 = ∫ 𝑢2 𝑑𝑢
1
= 𝑢3 + 𝑐
3
1
= 𝑠𝑖𝑛3 𝑥 + 𝑐
3
1
sin 𝑦 = 2 ( 𝑠𝑖𝑛3 𝑥 + 𝑐)
3
2
𝑦 = arcsin ( 𝑠𝑖𝑛3 𝑥 + 2𝑐) 𝑙𝑒𝑡 2𝑐 = 𝐴
3
𝟐
𝒚 = 𝐚𝐫𝐜𝐬𝐢𝐧 ( 𝒔𝒊𝒏𝟑 𝒙 + 𝑨)
𝟑
, QUESTION 2.
𝑑𝑦
(𝑥 3 + 𝑦 3 ) = 3𝑥𝑦 2
𝑑𝑥
Assignment 01 Answers
Semester 1 2025
0027 63 985 5033
,
,QUESTION 1.
𝑑𝑦
cos 𝑦 = 2 𝑠𝑖𝑛2 𝑥 cos 𝑥
𝑑𝑥
∫ cos 𝑦 𝑑𝑦 = 2 ∫ 𝑠𝑖𝑛2 𝑥 cos 𝑥 𝑑𝑥
sin 𝑦 = 2 ∫ 𝑠𝑖𝑛2 𝑥 cos 𝑥 𝑑𝑥
For the integral
∫ 𝑠𝑖𝑛2 𝑥 cos 𝑥 𝑑𝑥 𝑙𝑒𝑡 𝑢 = sin 𝑥 𝑑𝑢 = cos 𝑥
∫ 𝑠𝑖𝑛2 𝑥 cos 𝑥 𝑑𝑥 = ∫ 𝑢2 𝑑𝑢
1
= 𝑢3 + 𝑐
3
1
= 𝑠𝑖𝑛3 𝑥 + 𝑐
3
1
sin 𝑦 = 2 ( 𝑠𝑖𝑛3 𝑥 + 𝑐)
3
2
𝑦 = arcsin ( 𝑠𝑖𝑛3 𝑥 + 2𝑐) 𝑙𝑒𝑡 2𝑐 = 𝐴
3
𝟐
𝒚 = 𝐚𝐫𝐜𝐬𝐢𝐧 ( 𝒔𝒊𝒏𝟑 𝒙 + 𝑨)
𝟑
, QUESTION 2.
𝑑𝑦
(𝑥 3 + 𝑦 3 ) = 3𝑥𝑦 2
𝑑𝑥
