;;;;;
;;;;;
;;;;; Integration
;;;;;
;;;;;

(define $Sd
  (lambda [$x $f]
    (match f math-expr
      {; symbols
       [,x (* (/ 1 2) x^2)]
       [<symbol _> (* f x)]
       ; function application
       [(,exp ,x) (exp x)]
       [(,cos ,x) (sin x)]
       [(,sin ,x) (* -1 (cos x))]
       [(,log ,x) (multSd x 1 (log x))]
       [(,** $a ,x) (/ (** a x) (log a))]
       [(,** $a $y) (with-symbols {t}
                      (substitute {[t y]} (Sd t (* (** a t) (d/d (inverse t y x) t)))))]
       [(,Sd $y $g) (`Sd x (`Sd y g))]
       [($f $y) (with-symbols {t}
                  (substitute {[t y]} (Sd t (* (f t) (d/d (inverse t y x) t)))))]
       ; term (constant)
       [,0 0]
       [<term $c <nil>> (* c x)]
       ; term (multiplication)
       [<mult $a <ncons $n ,x $r>>
        (if (contain-symbol? x r)
          (`Sd x f)
          (* (/ a (+ n 1)) (** x (+ n 1)) r))]
       ; polynomial
       [<poly $ts> (sum (map (Sd x $) ts))]
       ; quotient
       [<div <plus $ts> $p2>
        (sum (map 1#(Sd x (/ %1 p2)) ts))]
       [<div $p1 $p2>
        (if (contain-symbol? x p2)
          (`Sd x f)
          (/ (Sd x p1) p2))]
       })))

(define $multSd
  (lambda [$x $f $g]
    (let {[$F (Sd x f)]}
      (- (* F g)
         (Sd x (* F (d/d g x)))))))

(define $dSd
  (lambda [$x $a $b $f]
    (let {[$F (Sd x f)]}
      (- (substitute {[x b]} F)
         (substitute {[x a]} F)))))