(define \$f (lambda [\$x] x)) (define \$multSd (lambda [\$x \$f \$G] (let {[\$F (Sd x f)]} (- (* F G) (Sd x (* f (d/d G x))))))) (multSd x (cos x) (f x));(+ (* (sin x) x) (* -1 (sin x))) (multSd x (cos (* 2 x)) (f x));(/ (+ (* 2 (sin (* 2 x)) x) (* -2 (sin (* 2 x)))) 4) (multSd x (cos (* n x)) (f x));(/ (+ (* (sin (* n x)) x n) (* -1 (sin (* n x)) n)) n^2) (multSd x (sin x) (f x));(+ (* -1 (cos x) x) (cos x)) (multSd x (sin (* 2 x)) (f x));(/ (+ (* -1 (cos (* 2 x)) x) (cos (* 2 x))) 2) (multSd x (sin (* n x)) (f x));(/ (+ (* -1 (cos (* n x)) x) (cos (* n x))) n) (define \$as (map (lambda [\$n] (let {[\$F (multSd x (cos (* n x)) (f x))]} (/ (- (substitute {[x π]} F) (substitute {[x (* -1 π)]} F)) π))) nats)) (take 10 as) ;{0 0 0 0 0 0 0 0 0 0} (define \$bs (map (lambda [\$n] (let {[\$F (multSd x (sin (* n x)) (f x))]} (/ (- (substitute {[x π]} F) (substitute {[x (* -1 π)]} F)) π))) (take 10 nats))) (take 10 bs) ;{2 -1 (/ 2 3) (/ -1 2) (/ 2 5) (/ -1 3) (/ 2 7) (/ -1 4) (/ 2 9) (/ -1 5)} (define \$f' (map (lambda [\$k \$b] (* b (sin (* k x)))) (zip nats bs))) (take 10 f') ;{(* 2 (sin x)) (* -1 (sin (* 2 x))) (/ (* 2 (sin (* 3 x))) 3) (/ (* -1 (sin (* 4 x))) 2) (/ (* 2 (sin (* 5 x))) 5) (/ (* -1 (sin (* 6 x))) 3) (/ (* 2 (sin (* 7 x))) 7) (/ (* -1 (sin (* 8 x))) 4) (/ (* 2 (sin (* 9 x))) 9) (/ (* -1 (sin (* 10 x))) 5)} (take 10 (map (substitute {[x (/ π 2)]} \$) f')) ;{2 0 (/ -2 3) 0 (/ 2 5) 0 (/ -2 7) 0 (/ 2 9) 0} ; = (/ pi 2) (map (/ \$ 2) (take 10 (map (substitute {[x (/ π 2)]} \$) f'))) ;{1 0 (/ -1 3) 0 (/ 1 5) 0 (/ -1 7) 0 (/ 1 9) 0} ; = (/ pi 4)