Q21 [CE AMath 8311 12] Let f(x) be a function of x and let k and s be constants. (a) By using the substitution y = x + ks, show that Hence show that, for any positive integer n, -(k+1)s [ f(x + ks)dx = √²+¹fC
Q21 [CE AMath 8311 12] Let f(x) be a function of x and let k and s be constants. (a) By using the substitution y = x + ks, show that Hence show that, for any positive integer n, -(k+1)s [ f(x + ks)dx = √²+¹fC
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Now Senior Secondary Mathematics (Module 2: Calculus and Algebra).
New Senior Secondary Mathematics (Module 2: Calculus and Algebra)
Advanced Exercise Ch. 10: Definite Integral
Q21 [CE AMath 8311 12]
Let f(x) be a function of x and let k and s be constants.
By using the substitution y = x + ks, show that
(a)
(k+1)s
["f(x + ks)dx= [(x+¹)*f f(x)dx.
Jks
Hence show that, for any positive integer n
[(x) + f(x + s) + + f(x + (n − 1)s)]dx = f(x)dx.
By using the substitution x = sin 8, evaluate
Using this result together with (a), evaluate
20
1
1
1
1
+
1-x²
++
dx.
* √₁-(x + +)* * √₁-(x + ²)²
√₁-(+2).
1-
1-
1-
Q22 [CE AMath 80II 12]
(a) Given that f(x) = f(a-x) for all real values of x, by using the substitution u = a-x, show that
["xf(x) dx = a f" f(u)du - ["uf(u) du
Hence deduce that
["xf(x) dx = f(x)dx.
(b) By using the substitution u = x-
sin¹ x
cos* u
-dx =
sin¹ x + cos x
sin u + cos¹ u
0
By using this result and
["f(x)dx = [ f(x)dx + ["f(x)dx,
evaluate
sin¹ x
S
-dx
sin¹ x + cos¹ x
(c) Using (a) and (b), evaluate
x sin¹ x
S
sin¹ x + cos4 x
Answer
(b)
=x-show that
f
-dx.
58
- (x + ²²-1) ²
du.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4ae89d13-3da3-4f42-8a30-4752fc9f6b5c%2Fa12c9bd2-8a5c-457e-8613-2bb606ea7556%2Fpq3oafs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Now Senior Secondary Mathematics (Module 2: Calculus and Algebra).
New Senior Secondary Mathematics (Module 2: Calculus and Algebra)
Advanced Exercise Ch. 10: Definite Integral
Q21 [CE AMath 8311 12]
Let f(x) be a function of x and let k and s be constants.
By using the substitution y = x + ks, show that
(a)
(k+1)s
["f(x + ks)dx= [(x+¹)*f f(x)dx.
Jks
Hence show that, for any positive integer n
[(x) + f(x + s) + + f(x + (n − 1)s)]dx = f(x)dx.
By using the substitution x = sin 8, evaluate
Using this result together with (a), evaluate
20
1
1
1
1
+
1-x²
++
dx.
* √₁-(x + +)* * √₁-(x + ²)²
√₁-(+2).
1-
1-
1-
Q22 [CE AMath 80II 12]
(a) Given that f(x) = f(a-x) for all real values of x, by using the substitution u = a-x, show that
["xf(x) dx = a f" f(u)du - ["uf(u) du
Hence deduce that
["xf(x) dx = f(x)dx.
(b) By using the substitution u = x-
sin¹ x
cos* u
-dx =
sin¹ x + cos x
sin u + cos¹ u
0
By using this result and
["f(x)dx = [ f(x)dx + ["f(x)dx,
evaluate
sin¹ x
S
-dx
sin¹ x + cos¹ x
(c) Using (a) and (b), evaluate
x sin¹ x
S
sin¹ x + cos4 x
Answer
(b)
=x-show that
f
-dx.
58
- (x + ²²-1) ²
du.
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