MAT 272 Test 1 – Spring 2019

MAT 272 Test 1 – Spring 2019 Name: KEY
There are 100 minutes for the entire test. A scientific calculator is allowed. You must show your integrals, work, etc.
for credit.
1. Evaluate the following integrals. (48 points)
2x
a)  ( x  2)( x  1)dx
2𝑥 𝐴 𝐵
= + → 2𝑥 = 𝐴(𝑥 + 1) + 𝐵(𝑥 + 2)
(𝑥 + 2)(𝑥 + 1) 𝑥 + 2 𝑥 + 1
𝑥 = −2 → 𝑨 = 𝟒; 𝑥 = −1 → 𝑩 = −𝟐
4 2 𝑑𝑥 𝑑𝑥
= ∫( +− ) 𝑑𝑥 = 4 ∫ − 2∫
𝑥+2 𝑥+1 𝑥+2 𝑥+1
= 𝟒 𝐥𝐧|𝒙 + 𝟐| − 𝟐 𝐥𝐧|𝒙 + 𝟏| + 𝑪

 x sec ( x) dx
2
b)
𝑢 = 𝑥 → 𝑑𝑢 = 𝑑𝑥; 𝑑𝑣 = sec 2 𝑥 𝑑𝑥 → 𝑣 = tan 𝑥
𝑢𝑣 − ∫ 𝑣𝑑𝑢 = 𝑥 tan 𝑥 − ∫ tan 𝑥 𝑑𝑥
= 𝒙 𝐭𝐚𝐧 𝒙 − 𝐥𝐧|𝐬𝐞𝐜 𝒙| + 𝑪



 cos (5x) dx
2
c)
1 1
cos 2 𝑥 = (1 + cos(2𝑥)) → ∫(1 + cos(10𝑥))𝑑𝑥
2 2
1 1
= ∫ 𝑑𝑥 + ∫ cos(10𝑥) 𝑑𝑥 ; 𝑢 = 10𝑥 → 𝑑𝑢 = 10𝑑𝑥
2 2
1 1 𝑑𝑢 1 1 1 1
= 𝑥 + ∫ cos 𝑢 ( ) = 𝑥 + ∫ cos 𝑢 𝑑𝑢 = 𝑥 + sin 𝑢 + 𝐶
2 2 10 2 20 2 20
𝟏 𝟏
= 𝒙+ 𝐬𝐢𝐧(𝟏𝟎𝒙) + 𝑪
𝟐 𝟐𝟎

1
d) x 2
9  x2
dx

1
𝑥 = 3 sin 𝜃 → 𝑑𝑥 = 3 cos 𝜃 𝑑𝜃 → ∫ 3 cos 𝜃 𝑑𝜃
9 sin2 𝜃 (3 cos 𝜃)

1 1 1 1
= ∫ 2 𝑑𝜃 = ∫ csc 2 𝜃 𝑑𝜃 = − cot 𝜃 + 𝐶
9 sin 𝜃 9 9
𝑥
𝑥 = 3 sin 𝜃 → sin 𝜃 = 3 → cot 𝜃 + 𝐶

𝟏 √𝟗 − 𝒙𝟐
=− ⋅ +𝑪
𝟗 𝒙

−√𝟗 − 𝒙𝟐
= +𝑪
𝟗𝒙




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