MTH 1730
Worksheet (11/20/08)
“Integration”
(1) Calculate the following integrals.
(a) Z2x−5
x2−5xdx
(b) Z2x−5
(x2−5x)4dx
(c) Z√x(1 + 2x)dx
(d) Z1
0
ex√ex+ 1 dx
(e) Z5
3
3x
√x−2dx (Use the substitution u=x−2.)
(f) Zxarctan x dx (Use integration by parts.)
(2) The rate of infection of a disease (in people per month) is given by the function
I0(t) = 100t
t2+ 1, where tis the time in months since the disease broke out. Find the
total number of infected people during the first four months of the disease.
(3) Suppose the temperature in a river at a point xin meters downstream from a factory
that is discharging hot water into the river is given by T(x) = 400 −0.25x2. Find
the average temperature over the interval [10,40].
(4) A region Ris enclosed by the curves y=xand y=x2. Find the area of R.
(5) Does the improper integral Z∞
1
6e−xdx converge or diverge? If it converges, what
does it converge to?