262
CHAPTER 32
32.16
Evaluate
32.17
Evaluate
Then
Let
32.19
Evaluate
Let x = f
2 . Then
[by Problem 32.6].
32.20 Evaluate
By Problem 28.1,
[Here, we used L'Hopital's rule to see that
32.18
Evaluate
Let
Then
[by Problem 32.6].
32.21
Find
By the reduction formula of Problem 28.42 and the result of Problem 32.20,
So,
32.22 Show that
for all natural numbers n.
By Problem 32.14, we know that the formula holds for « = 0. Assume now, for the sake of induction, that
the formula holds for n — 1. By the reduction formula of Problem 28.42,
x"~
l e~* dx = n • (n — 1)! = nl. [The gamma function T(u) is defined as
problem shows that F(n + !) = «!.]
This
32.23 Investigate
Thus, the integral diverges.
32.24
Investigate
32.25 Investigate
Thus, the integral diverges.
2u du = dx.
xV* dx.
2)-2]} = 2.
xV
x dx.
xV* dx = -xV* +
x
2 e~'dx = 0 + 3-2 = 3!.
x
3 e~* dx = lira (-*V*) ]"„ + lim 3
U-» + "
u-» + oo
3 J x V* dx.
0°°x"e * dx = n\
n