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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
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.
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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