7.4. EXERCISES 187

18. Define F (x) ≡∫ x

01

1+t2 dt. Of course F (x) = arctan(x) as mentioned above but justconsider this function in terms of the integral. Sketch the graph of F using only itsdefinition as an integral. Show there exists a constant M such that −M ≤ F (x)≤ M.Next explain why limx→∞ F (x) exists and show this limit equals − limx→−∞ F (x).

19. In Problem 18 let the limit defined there be denoted by π/2 and define T (x) ≡F−1 (x) for x ∈ (−π/2,π/2) . Show T ′ (x) = 1+ T (x)2 and T (0) = 0. As part of

this, you must explain why T ′ (x) exists. For x ∈ [0,π/2] let C (x) ≡ 1/√

1+T (x)2

with C (π/2) = 0 and on [0,π/2] , define S (x) by√

1−C (x)2. Show both S (x) andC (x) are differentiable on [0,π/2] and satisfy S′ (x) =C (x) and C′ (x) =−S (x) . Findthe appropriate way to define S (x) and C (x) on all of [0,2π] in order that these func-tions will be sin(x) and cos(x) and then extend to make the result periodic of period2π on all of R. Note this is a way to define the trig. functions which is independentof plane geometry and also does not use power series. See the book by Hardy (If Iremember correctly), [19] for this approach.

20. Let p,q > 1 and satisfy 1p +

1q = 1. Let x = t p−1. Then solving for t, you get t =

x1/(p−1) = xq−1. Explain this. Now let a,b ≥ 0. Sketch a picture to show why∫ b

0xq−1dx+

∫ a

0t p−1dt ≥ ab.

Now do the integrals to obtain a very important inequality bq

q + ap

p ≥ ab. When willequality hold in this inequality?

21. Suppose f ,g are two Riemann integrable functions on [a,b]. Verify Holder’s inequal-ity. ∫ b

a| f | |g|dx ≤

(∫ b

a| f |p dx

)1/p(∫ b

a|g|q dx

)1/q

Hint: Do the following. Let A =(∫ b

a | f |p dx)1/p

,B =(∫ b

a |g|q dx)1/q

. Then let

a = | f |A ,b = |g|

B and use the wonderful inequality of Problem 20.

22. If F,G are antiderivatives for f ,g on R, show F (x) = G(x)+C for some constant,C. Use this to give a proof of the fundamental theorem of calculus which has for itsconclusion

∫ ba f (t)dt = G(b)−G(a) where G′ (x) = f (x) . Use the version of the

fundamental theorem of calculus which says that (∫ x

a f (t)dt)′ = f (x) for f continu-ous.

23. Suppose f and g are continuous functions on [a,b] and that g(x) ̸= 0 on (a,b) . Showthere exists c ∈ [a,b] such that f (c)

∫ ba g(x) dx =

∫ ba f (x)g(x) dx. Hint: Define m ≡

min{ f (x) : x ∈ [a,b]} ,M ≡ max{ f (x) : x ∈ [a,b]}. Now consider∫ ba f (x)g(x) dx∫ b

a g(x) dxor∫ b

a f (x)(−g(x)) dx∫ ba (−g(x)) dx

Argue that one of these quotients is between m and M. Use intermediate value theo-rem.