Solution of QUIZ #1
∫2
1. Decide whether the integral −2 ln(x2 ) dx is improper. Explain your reasoning.
solution: ∫
2
The integral −2 ln(x2 ) dx is improper.
Because the function has an infinite discontinuity in [−2, 2] at x = 0.
(10%)
2. Determine whether
diverges or
∫ ∞ the improper∫ integral
∫ ∞converges. Evaluate the integral if it
∞
converges. (a) 0 e−x/3 dx, (b) e lnxx dx and (c) −∞ x2x+1 dx.
(45%)
solution:
(a)
∫ b
lim
e−x/3 dx = lim [−3e−x/3 ]b0 = lim −3e−b/3 + 3 = 3(converges)
b→∞
(b)
b→∞
0
∫
b
lim
b→∞
(c)
∫∞
e
b→∞
ln x
(ln x)2 b
(ln b)2
1
dx = lim [
]e = lim [
− ] = ∞(diverges)
b→∞
b→∞
x
2
2
2
∫0
∫∞
dx = −∞ x2x+1 dx + 0 x2x+1 dx(consider the one side)
ln(x2 + 1) b
ln(b2 + 1)
x
dx
=
lim
[
]
=
lim
[
] = ∞(diverges)
0
b→∞
b→∞
x2 + 1
2
2
x
−∞ x2 +1
∫ b
lim
b→∞
0
3. A business is expected to yield a continuous flow of profit at the rate of 5% compounded
continuously, what is the present value of the business (a) for 10 years? (b) forever? (20%)
solution:
(a)
∫1
0f e−0.05t dt = [−20f e−0.05t ]10 0 = −20f e−0.5 + 20f
0
(b)
∫ ∞ −0.05t
fe
dt = [−20f e−0.05t ]∞
0 = 20f
0
∫∞
4. Show that the improper integral 0 xe−x dx converges, then evaluate the integral.
(25%)
solution:
∫
b
lim
b→∞
xe−x dx
0
let u = x, du∫= 1, dv = e−x , v = −e−x
lim [xe−x − −e−x dx]b0 = lim [xe−x − e−x ]b0 = lim [(be−b − e−b ) − (−1)] = 1
b→∞
b→∞
b→∞