
Prove the following integral. ., xPn-1(x)P,(x)dx = 2n (2n-1)(2+1) 2 use (n + 1)Pn+1(x) – (2n...
--Red In( - ) () -2 21 (+1) 2+1(n + m)} ml(m + 2n + 1) (1 - 1) de 22+1(n!) (2n +1) 8. By evaluating ac 2h +G ah where G(h) is the generating function for Legendre polynomials, show that 1 - 2 Σ (2n +1) Po (1 - 2ch + ha) Hence, or otherwise, prove that Pn(x) dx 2h 9. Given that {(2, 2) = ( 12h the hm-dh m>1. prove that 2am+ | 112,0)P.a)dt (m + n)...
(d) Use the Integral Inequality to prove that sin (rx(In(x + 1) dx = 1 671 (2) - In(3) - ;) 4
2. Suppose P and Q are positive odd integers such that (PQ)-1. Prove that Qm] Pn] P-1 0-1 0<m<P/2 0<n
Find dx for the following integral. 0 dt } 1+P tan x dy dx (Simplify your answer.)
Prove the recurrence relation
xPn(x) = n + 1
2n + 1
Pn+1(x) + n
2n + 1
Pn−1(x),
and evaluate the following integral using the orthogonal property of Legendre polynomials
Z 1
−1
xPn(x)Pn−1(x)dx
Problem 9: Prove that 1/2n) s [1 x 3x5x. .. x (2n is a positive integer 1)/2 x4x. ..x 2n) whenever n
3 11 Use Simpson's rule with n=1 (so there are 2n = 2 subintervals) to approximate dx. 1 + x2 The approximate value of the integral from Simpson's rule is (Round the final answer to two decimal places as needed. Round all intermediate values to four decimal places as needed.) 5 Use Simpson's rule with n=4 (so there are 2n = 8 subintervals) to approximate OX dx and use the fundamental theorem of calculus to find the exact value of...
use mathematical induction to prove the following
* n(n+1)(n+2) 34 + 1) = n(n + y(n = 3). 2* = 2n+1 – 1. (4k + 1) = (n + 1)(2n + 1). k=0
Use the Fundamental Theorem of Calculus to evaluate the following definite integral. 2 6 dx S √1-x² 0 V3 2 6 dx 5 0 V1 - (Type an exact answer.)
(2) Let Pn [x] = {p € P[x] : degp <n}, where P[x] is the set of all polynomials. Let the polynomials li() defined by II;tilt - a;) i=0,1,...11 bi(T) = 11: a; - aj) where aj, j = 0,1,..., are distinct real numbers and aia . Show that (d) The change of basis transformation from the standard basis ', j = 0,1,...,n to l; () is given by the Vandermonde matrix (1 00 ... am 1 01 .01 1...