Edy Historia

Challenge Problems
y x

495

a 2 y f(x)

its mirror image with respect to the line x a>2 has x-coordinate x¿ a x. Therefore, f(a x¿) f(a (a x)) f(x) . (See Figure 1.) Since congruent figures have equal areas, the result follows from interpreting definite integrals as areas. b. Using the result of part (a), we see that
p>2 p>2

y 0

f(a

x)
0

sinm x dx
0

sinm a asin

p 2xb dx cos
m p sin xb dx 2

x

x

x
0

p>2

FIGURE 1 The graphs of f(x) and f(a x) are mirror images with respect to x a>2.

p cos x 2

p>2

cosm x dx
0

c. Using the result of part (b) with m
p>2

2, we have
p>2

I
0

sin2 x dx
0

cos2 x dx

Therefore,
p>2 p>2

2I
0

sin2 x dx
0 p>2

cos2 x dx
p>2

(sin2 x
0

cos2 x) dx
0

dx

p 2

and henceI

p>4.

CHALLENGE PROBLEMS
x dx, where a b. x x 12 Œx œ 1 Œx œ 2. Show that Œ tœ dt 2 0 Œ xœ is the greatest integer function. 1. Evaluate
a 10p b

8. Evaluate Œxœ 1x Œx œ 2 , where
x3

tan t 1>3 dt lim
x→0
0 0 2x2

3. Evaluate
0

11

t dt 9. Evaluate lim
b→a

cos 2x dx. 1 b
n 13>2

b

4. By interpreting the integral geometrically, evaluate
12>2 1

21

a

f(x)dx, where f is a continuous
a

function. x 2 dx
3

10. Evaluate lim a 2 n→ k 1 n 4x x2 3x
2

n k2

.

5. Evaluate
1

3x

6

2x

5

5x

1

dx.

Hint: Relate the limit to the limit of a Riemann sum of an appropriate function.
b 11. a. Show that ab f(x) dx f(a b x) dx, and a give a geometric interpretation of the result. b. Use the result of part (a) to show that p f(sinx) cos x dx 0. 0

6. Find 7. Find

1 dx. sin x cos4 x
2

1

dx 2x cos a

x2

, where 0
x cos a . sin a

a

p.

12. Show that

Hint: Use the substitution u

t 0

f(x)t(t

x) dx

t 0

t(x)f(t

x) dx.

496

Chapter 4 Integration b. Use the result of part (a) to evaluate
4 2>3

13. a. Suppose that f is continuous and t and h are differentiable. Show that d dxh(x)

cos(x f(t)dt f [h(x)]h¿(x)
1x

4)2 dx

3
1>3

cosc9ax

f [t(x)]t¿(x)

3

2 2 b d dx 3

t (x)

b. Use the result of part (a) to find t¿(x) if t(x)
1>x

19. Suppose that f is a continuous periodic function with period p. a. Prove that if a is any real number, then
a a p

sin t 2 dt

x

0
0

f(x) dx
p

f(x) dx

14. Prove that if f and t are continuousfunctions on [a, b], then `
b a

f(x)t(x) dx `

B

b

b

[ f(x)]2 dx
a a

[t(x)]2 dx

b. Use the result of part (a) to show that if a is any real number, then
p a p

This is known as Schwarz’s inequality.
Hint: Consider the function F(x)
number. [ f(x) tt(x)]2, where t is a real
0

f(x) dx
a

f(x) dx

15. a. Use Schwarz’s inequality (see Exercise 14) to prove that
1 021

x dx

3

15 2

20. Let f be continuous on an interval [ a, a]. a. Show that a a f(x 2) dx 2 0a f(x 2) dx. b. What can you say about
a a

f(x 2)sin x dx? 0

b. Is this estimate better than the one obtained by using the Mean Value Theorem for Integrals? 16. Find the values of x at which
x2 2

21. Let f be continuous on an interval [a, b] and satisfy x b f(t) dt f(t) dt for all xin [a, b]. Show that f(x) a x on [a, b]. 22. The Fresnel function S is defined by the integral
x

F(x)
0

t

5t t2 1

4

S(x) dt
0

sina

pt 2 b dt 2

has relative extrema. 17. Suppose that f is continuous on an interval [a, b]. Show that 1 n lim a f ca n→ n k 1 18. a. Prove that
b 1

a. Sketch the graphs of f(x) sin(px 2>2) and S(x) on the same set of axes for 0 x 3.Interpret your results. b. Sketch the graph of S on the interval [ 10, 10]. 23. Find all continuous, nonnegative functions f defined on [0, b], where b 0, satisfying the equation [ f(x)]2 2 0x f(t) dt. 24. a. Prove that e R sin x where R 0. e(
2R>p)x

k(b n

a)

d

1 b a
a

b

f(x) dx

f(x) dx
a

(b

a)
0

f [(b

a)t

a] dt

Hint: Show that f(x)
p>2

b. Use the result of...