Rob Bayer Math 1B PDP Worksheet February 12, 2009
The Limit Comparison Test
1. Determine whether each of the following integrals converges or diverges:
(a) Z∞
10
x4
−3x2+ 10
x5+ 3x−2dx
(b) Z1
0
x2+ 3√x+3
√x
x2/3+x2−x3dx
(c) Z∞
1
xsin 1
x2dx
(d) Z∞
2
sin( 1
x)
ln(x+ 1) −ln xdx
(e) Z1
0
dx
xtan−1x
(f) Z∞
3
2x2+ 3x
√1 + x5
dx
2. Determine whether the integral Z∞
1
sin2x
x√xconverges or diverges by using
(a) The Limit Comparison Test
(b) The regular Comparison Test
(c) Which is easier for problems like this?
Integration Practice
1. Find each of the following integrals. Be sure to work as a group so everyone knows how to do all these
problems.
(a) Zex+ex
dx
(b) Zsec2(sin θ)
sec θdθ
(c) Z1
√x+1+√x
dx
(d) Zln(x+ 1)
x2dx
(e) Zt3+ 1
t3−t2dt
(f) Zcos4t−sin4tdt
(g) Zcos32xsin 2xdx
(h) Zdt
√et
(i) Z1
x√x2+ 4
dx
(j) Z1
x√x+ 4dx
(k) Zx
√x2+ 4
dx
(l) Z1
4
√x+3
√x
(m) Zln(sec θ) sec2θdθ