MAT1512 Assignment 7 2021

MAT1512 ASSIGNMENT 7 2021



Question 1



(𝑎)


2
𝐼 = ∫ (𝑈 6 − 2𝑈 5 + ) 𝑑𝑈
7
2
𝐼 = ∫ (𝑈 6 − 2𝑈 5 + 𝑈 0 ) 𝑑𝑈
7
1 2 2
𝐼= 𝑈 6+1 − 𝑈 5+1 + 𝑈 0+1 + 𝑐
6+1 5+1 7 × (0 + 1)
1 2 2
𝐼 = 𝑈7 − 𝑈6 + 𝑈1 + 𝑐
7 6 7×1
1 1 2
𝐼 = 𝑈7 − 𝑈6 + 𝑈 + 𝑐
7 3 7


(𝑏)



1 + √𝑥 + 𝑥
𝐼 = ∫( ) 𝑑𝑥
𝑥

1 √𝑥 𝑥
𝐼 = ∫( + + ) 𝑑𝑥
𝑥 𝑥 𝑥
1
1 𝑥2
𝐼 = ∫ ( + + 1) 𝑑𝑥
𝑥 𝑥

1
1 𝑥2
𝐼 = ∫ ( + 1 + 1) 𝑑𝑥
𝑥 𝑥

1 1
𝐼 = ∫ ( + 𝑥 2−1 + 1) 𝑑𝑥
𝑥
1 1 2
𝐼 = ∫ ( + 𝑥 2−2 + 1) 𝑑𝑥
𝑥
1 1−2
𝐼 = ∫ ( + 𝑥 2 + 1) 𝑑𝑥
𝑥

, 1 1
𝐼 = ∫ ( + 𝑥 −2 + 1𝑥 0 ) 𝑑𝑥
𝑥
1 1 1
𝐼 = 𝑙𝑛|𝑥| + 𝑥 −2+1 + 𝑥 0+1 + 𝑐
1 0 + 1
−2 + 1

1 1 1 1
𝐼 = 𝑙𝑛|𝑥| + 𝑥2 + 𝑥 + 𝑐
1 1
2
1
𝐼 = 𝑙𝑛|𝑥| + 2𝑥 2 + 1𝑥 + 𝑐

𝐼 = 𝑙𝑛|𝑥| + 2√𝑥 + 𝑥 + 𝑐


(𝑐)


4
4 + 6𝑢
𝐼 = ∫( ) 𝑑𝑢
√𝑢
1
4
4 + 6𝑢
𝐼 = ∫( 1 ) 𝑑𝑢
1 𝑢2
4
4 6𝑢
𝐼 = ∫ ( 1 + 1 ) 𝑑𝑢
1 𝑢2 𝑢2
4
1 1
𝐼 = ∫ (4𝑢−2 + 6𝑢1 𝑢−2 ) 𝑑𝑢
1
4
1 1
𝐼 = ∫ (4𝑢−2 + 6𝑢1−2 ) 𝑑𝑢
1
4
1 1
𝐼 = ∫ (4𝑢−2 + 6𝑢2 ) 𝑑𝑢
1


4 1 6 1
𝑢=4
𝐼= 𝑢−2+1 + 𝑢2+1 ]
1 1 𝑢=1
− +1 +1
2 2

4 1 6 1 2
𝑢=4
𝐼= 𝑢2 + 𝑢 2+2 ]
1 1 2 𝑢=1
2 2+2

4 1 6 1+2
𝑢=4
𝐼= 𝑢2 + 𝑢 2 ]
1 1+2 𝑢=1
2 2