204 CHAPTER 7. THE DERIVATIVE

14. I found this problem in Apostol’s book [1]. This is a very important result and is ob-

tained very simply. Read it and fill in any missing details. Let g(x)≡∫ 1

0e−x2(1+t2)

1+t2 dt

and f (x)≡(∫ x

0 e−t2dt)2

. Note ∂

∂x

(e−x2(1+t2)

1+t2

)=−2xe−x2(1+t2). Explain why this

is so. Also show the conditions of Problem 13 are satisfied so that

g′ (x) =∫ 1

0

(−2xe−x2(1+t2)

)dt.

Now use the chain rule and the fundamental theorem of calculus to find f ′ (x) . Thenchange the variable in the formula for f ′ (x) to make it an integral from 0 to 1 andshow f ′ (x)+g′ (x) = 0. Now this shows f (x)+g(x) is a constant. Show the constantis π/4 by letting x→ 0. Next take a limit as x→ ∞ to obtain the following formula

for the improper integral,∫

∞

0 e−t2dt,(∫

∞

0 e−t2dt)2

= π/4. In passing to the limit inthe integral for g as x→ ∞ you need to justify why that integral converges to 0. Todo this, argue the integrand converges uniformly to 0 for t ∈ [0,1] and then explainwhy this gives convergence of the integral. Thus

∫∞

0 e−t2dt =

√π/2.

15. Recall the treatment of integrals of continuous functions in Proposition 5.9.5 or whatyou used in beginning calculus. The gamma function is defined for x > 0 as Γ(x)≡∫

∞

0 e−ttx−1dt ≡ limR→∞

∫ R0 e−ttx−1dt. Show this limit exists. Note you might have to

give a meaning to∫ R

0 e−ttx−1dt if x < 1. Also show that Γ(x+1) = xΓ(x) , Γ(1) = 1.How does Γ(n) for n an integer compare with (n−1)!?

16. Show the mean value theorem for integrals. Suppose f ∈C ([a,b]) . Then there existsx ∈ (a,b), not just in [a,b] such that f (x)(b−a) =

∫ ba f (t)dt. Hint: Let F (x) ≡∫ x

a f (t)dt and use the mean value theorem, Theorem 5.9.3 along with F ′ (x) = f (x).

17. Show, using the Weierstrass approximation theorem that linear combinations of theform ∑i, j ai jgi (s)h j (t) where gi,h j are continuous functions on [0,b] are dense inC ([0,b]× [0,b]) , the continuous functions defined on [0,b]× [0,b] with norm givenby

∥ f∥ ≡max{| f (x,y)| : (x,y) ∈ [0,b]× [0,b]}

Show that for h,g continuous,∫ b

0∫ s

0 g(s)h(t)dtds−∫ b

0∫ b

t g(s)h(t)dsdt = 0. Nowexplain why if f is in C ([0,b]× [0,b]) ,∫ b

0

∫ s

0f (s, t)dtds−

∫ b

0

∫ b

tf (s, t)dsdt = 0.

18. Let f (x)≡(∫ x

0 e−t2dt)2

. Use Proposition 5.9.5 which includes the fundamental the-orem of calculus and elementary change of variables, show that

f ′ (x) = 2e−x2(∫ x

0e−t2

dt)= 2e−x2

(∫ 1

0e−(xs)2

xds)=∫ 1

02xe−x2(1+s2)ds.

Now show

f (x) =∫ 1

0

∫ x

02te−t2(1+s2)dtds.

Show limx→∞

∫ x0 e−t2

dt = 12√

π