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Name ______________________________
MA221 Calculus II
Chapter 5
Step-by-step work must be shown for every problem. No credit will be given for answers
that are not supported by detailed work.
#1
#2
2 /3 2sin
0
3
+ 2sin tan 2
d
sec2
0 6 dx
x
#3
0 t sin3tdt
#4
sin
#5
Determine whether the integral is convergent or divergent
#6
Determine whether the integral is convergent or divergent
Extra
3 x cos2 xdx
x3 + 4
dx
x2 + 4
1
33
0
x +1
dx .
x + 2x
2
( x −1)−1/5dx .
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Explanation & Answer
Tell your fr!end he owes me $5 😂😂😂See attached for my answers to this assignment.Let me know if you have any questions or if you have difficulty accessing the files.
Name ______________________________
MA221 Calculus II
Chapter 5
Step-by-step work must be shown for every problem. No credit will be given for answers
that are not supported by detailed work.
#1
2 /3 2sin
0
2sin tan 2
d
sec2
2 sin 𝜃 + 2 sin 𝜃 tan2 𝜃 2 sin 𝜃 (1 + tan2 𝜃)
𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦:
=
sec 2 𝜃
sec 2 𝜃
𝑈𝑠𝑒 𝑇𝑟𝑖𝑔 𝐼𝑑𝑒𝑛𝑡𝑖𝑡𝑦: 1 + tan2 𝑥 = sec 2 𝑥
→
2 sin 𝜃 (1 + tan2 𝜃 ) (2 sin 𝜃 )(sec 2 𝜃 )
=
= 2 sin 𝜃
sec 2 𝜃
sec 2 𝜃
2𝜋
3 2 sin 𝜃
∫
0
2𝜋
2𝜋
3
+ 2 sin 𝜃 tan2 𝜃
2𝜋
𝑑𝜃 = ∫ 2 sin 𝜃 𝑑𝜃 = −2 cos 𝜃 | 3 = −2 (cos ( ) − cos 0)
2
sec 𝜃
3
0
0
1
= −2 (− − 1) = 3
2
#2
3
0 6 dx
x
𝑎𝑥
Using the law of integration of exponential functions: ∫ 𝑎 𝑥 𝑑𝑥 = ln 𝑎 + 𝐶
3
∫ 6𝑥 𝑑𝑥 =
0
#3
6𝑥 3
1
1
215
(63 − 60 ) =
(216 − 1) =
| =
ln 6
ln 6
ln 6
ln 6
0
0 t sin3tdt
𝑢 = 𝑡 → 𝑑𝑢 = 𝑑𝑡
cos(3𝑡)
𝐿𝑒𝑡 {
𝑑𝑣 = sin(3𝑡) 𝑑𝑡 → 𝑣 = −
3
∫ 𝑢𝑑𝑣 = 𝑢𝑣 − ∫ 𝑣𝑑𝑢 = (𝑡) (−
=−
3𝑡 cos(3𝑡) sin(3𝑡)
+
3
9
cos(3𝑡)
1
𝑡 cos(3𝑡) 1 sin(3𝑡)
) − (− ) ∫ cos(3𝑡) 𝑑𝑡 = −
+ (
)
3
3
3
3
3
−𝑡𝑐𝑜𝑠(3𝑡) sin(3𝑡)
+
3
9
→ ∫ 𝑡 sin(3𝑡) 𝑑𝑡 =
𝜋
−𝜋 cos(3𝜋) sin(3𝜋)
−(0) ∙ cos(0) sin(0)
𝜋
𝜋
∫ 𝑡 sin(3𝑡) 𝑑𝑡 = [
]−[
]= −0=
+
+
3
9
3
9
3
3
0
#4
sin
3 x cos2 xdx
Use trig identify: sin2 𝑥 + cos 2 𝑥 = 1...
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