integrals are easy actually????

src/exercise/1.typ

@@ -109,6 +109,7 @@ $
 
 #set enum(numbering: "(a)")
 + $
+  f(x) &= ln((e^x) / (1+e^x)) \
   e(x) = e^x wide a(x) &= (e^x)/(1+e^x) wide b(x) = 1+e^x \
 
   ln: RR^+ |-> RR wide e&: RR |-> RR^+ wide => f: RR |-> RR <=> DD_f = RR \
@@ -124,3 +125,47 @@ $
   &= (1+e^x)/([1+e^x]^2) \
   &= 1/(1+e^x) checkmark
 $
++ $
+  D_f &= RR \\ {x = 1}\
+  f(x) &= ((x+1)/(x-1))^2 \
+  a(x) = x+1 wide b(x) = x-1 quad&quad c(x) = a(x)/b(x) wide d(x) = x^2 \
+  a'(x) = 1 wide b'(x) &= 1 wide d'(x) = 2x \
+  c'(x) &= (b' dot a - b dot a')/(a^2) \
+  &= (x-1 - (x+1))/((x-1)^2) \
+  &= (-2)/((x-1)^2) \
+  f'(x) &= d'(c(x)) dot c'(x) \
+  &= 2 dot (x+1)/(x-1) dot (-2)/((x-1)^2) \
+  &= -4 dot (x+1)/((x-1)^3) checkmark
+$
+
+#pagebreak()
+=== Exercise 4 @Exercise[1, 4]
+#block(
+  fill: luma(230),
+  inset: 8pt,
+  radius: 4pt,
+[
+  Calculate the following integrals using substitution:
+
+  #set enum(numbering: "(a)")
+  + $ integral (2x+7)/(x^2+7x+3) d x $
+  + $ integral (cos(ln(x)))/(x) d x $
+])
+
+#set enum(numbering: "(a)")
++ $
+  A &= integral (2x+7)/(x^2+7x+3) d x \
+  &= integral (a'(x))/(a(x)) d x \
+  &= integral (a'(x))/(a(x)) d x \
+  &= integral 1/(a(x)) dot a'(x) d x; wide d u = a'(x) d x \
+  &= integral 1/u d u \
+  &= ln(u) = ln(x^2+7x+3) checkmark
+$
++ $
+  A &= integral (cos(ln(x)))/(x) d x \
+  &= integral cos(ln(x)) dot 1/x d x \
+  &= integral cos(u) dot u' d x \
+  &= integral cos(u) dot d u \
+  &= sin(u) \
+  &= sin(ln(x)) \
+$