Question: Compute
R
x2
1−x2 dx.
Answer:
−x +
Question: Compute
R∞
0
1
1
ln |1 − x| + ln |1 + x| + C
2
2
xe−x dx.
Answer:
1
Question: Compute
R
1
x(1+(ln(x))2 ) dx.
Answer:
tan−1 (ln(x)) + C
Question: Find the length of the curve: y = ln(cos(θ)), 0 ≤ θ ≤ π/4.
Answer:
√
ln( 2 + 1)
Question: Determine whether the sequence an = ln(2 − n1 ) converges or diverges. If it converges, find its limit.
Answer:
Converges, limit = (ln(2))−
Question: Determine whether the sequence an =
find its limit.
n3
n2 +1
converges or diverges. If it converges,
Answer:
Diverges
Question: Determine whether the sequence an =
find its limit.
cos(n)
3n
converges or diverges. If it converges,
Answer:
Converges, limit = 0
Question: Determine whether the series
find its limit.
P∞
2
n=1 n(n+2)
1
converges or diverges. If it converges,
Answer:
Converges, limit =
Question: Determine whether the series
find its limit.
n2
n=1 1−2n2
P∞
3
2
converges or diverges. If it converges,
Answer:
Diverges
Question: Determine whether the series
its limit.
P∞
1
n=1 2n
converges or diverges. If it converges, find
Answer:
Converges, sum = 1
Question: Determine whether the sequence an =
find its limit.
3n−1
4n
converges or diverges. If it converges,
Answer:
Converges, limit = 0
Question: Determine whether the sequence an =
find its limit.
(−1)n
n!
converges or diverges. If it converges,
Answer:
Converges, limit = 0
Question: Compute
R
2
√x
dx
1−x2
converges or diverges. If it converges, find its limit.
Answer:
p
1
[sin−1 (x) − x 1 − x2 ] + C
2
Question: Determine whether the series
its limit.
P∞
n=0
2n +1
3n
converges or diverges. If it converges, find
Answer:
Converges, sum =
Question: Determine whether the series
its limit.
P∞
n=1
2
7 n
4
9
2
converges or diverges. If it converges, find
Answer:
Diverges
Question: Determine whether the series
its limit.
P∞
1
n=1 n2 +n
converges or diverges. If it converges, find
Answer:
Converges, sum = 1
Question: Compute
R
x2 +7x−16
x2 +6x−7 dx.
Answer:
x + 2 ln |x + 7| − ln |x − 1| + C
Question: Compute
R
1
x2 +4x+5 dx.
Answer:
tan−1 (x + 2) + C
Question: Find the exact area of the surface obtained by rotating the curve: y = 32 x2 , 1 ≤ x ≤ 2
around the y-axis.
Answer:
2π 3/2
[37 − 103/2 ]
27
3