(defn cube [n]
  (+ (* 5 n n n) (math/sin (* 2.5 n))))

(defn sum [term next a b]
  (if (> a b)
    0
    (+ (term a)
       (sum term next (next a) b))))

(defn integral-old [f a b dx]
  (defn add-dx [x]
    (+ x dx))
  (* (sum f add-dx (+ a (/ dx 2.0)) b) dx))

(defn integral [f a b n]
  (def h (/ (- b a) n))
  (defn incre [n]
    (+ n 1))
  (defn apply [k]
    (def multiplier
      (cond
        (or (= k 0) (= k n)) 1
        (= (mod k 2) 0) 2
        4))
    (* multiplier (f (+ a (* k h)))))
  (* (/ h 3) (sum apply incre 0 n)))

(defn integral-improved [f a b n]
  (def h (/ (- b a) n))
  (defn incre [k]
    (+ k 2))
  (defn apply [k]
    (f (+ a (* k h))))
  (* (/ h 3)
     (+
      (f a)
      (* 4 (sum apply incre 1 (- n 1)))
      (* 2 (sum apply incre 2 (- n 1)))
      (f b))))

(printf "%.23f" (integral-improved cube 0 1 10))
(printf "%.23f" (integral-improved cube 0 1 100))
(printf "%.23f" (integral-improved cube 0 1 1000))
(printf "%.23f" (integral-improved cube 0 1 10000))