If cos xdx = f(x) - 2x sin xdx, which of the followings can be the formula of the function f(x)? Sa - 12 Lütfen birini seçin: 2x Cos r?sin 2 sin + 2a cos (2_o?) cosa 4 sina sina 4 cos 2 sin
11) Find ? sin2 x cos2
xdx.
11) Find S sin? x cosa xdx.
QUESTION 17 Find the integral. - sin x cos xdx - sinox +C 6 - sinx 5 + sin 5x 5 +C sinx -C
Calculus II Question
Will give thumbs up! Don't know how to approach this.
(E) (sin xdx
For the set of functions {sin(x),sin(2x),sin(3x),...}=sin(nx)}, n=1,2,3,... on the interval [0,pi]. Show that the set of functions is orthogonal on [0,pi].
Consider the series following series of functions ' sin(nx) 3 n-1 a) Show that the series is absolutely and uniformly convergent on the real axis. Let f be its summation function n sin(nx) b) Show that f E C(R) and that 1 cos(nx) f'(x)= 2-1 c) Show that 「 f#072821) f(x)dx = k=0
Consider the series following series of functions ' sin(nx) 3 n-1 a) Show that the series is absolutely and uniformly convergent on the real axis. Let f...
3. Let W (x, t) = (coswt)(a cos nx+b sin nx) and (x, t) = (exp -kn-t)(a cos nx+ bsin nx). Here n is a positive integer, 2,t are real variables, and a, b,w, k are real constants with k positive. a. Evaluate W(x,0), H (2,0) and əW/ət(x,0) for all c. b. Show OH/ət = k(32H/8x2) for all x,t. c. Find some positive constant c so that w2w/at2 = c(32W/8x?) for all x, t.
7. Show that sin. nx-zs -zsin sin’x+ 1 4 3 sin' x sinº x +...= 2 sin x 2+sin x for all x ER
13. Evaluate, S sin 5x cos x dx. Also prove that, 52" sin mx cos nx dx = 0 Using reduction formula *****
*Prove that the orthogonal systems {sin nx)., and {ï, cos nxml are both complete on [0, T] 3.