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src/exercise/1.typ

@@ -169,3 +169,27 @@ $
   &= sin(u) \
   &= sin(ln(x)) \
 $
+
+#pagebreak()
+=== Exercise 5 @Exercise[1, 5]
+#block(
+  fill: luma(230),
+  inset: 8pt,
+  radius: 4pt,
+[
+  Calculate the following integrals using partial integration:
+
+  #set enum(numbering: "(a)")
+  + $ integral (x+2) dot e^(2x) d x $
+  + $ integral x^2 dot ln(x) d x $
+])
+
+#set enum(numbering: "(a)")
++ $
+  A &= integral (x+2) dot e^(2x) d x \
+  &= integral u(x) dot v'(x) d x \
+  &= u(x) dot v(x) - integral u'(x) dot v(x) d x \
+  &= (x+2) dot (e^(2x) dot 1/2x) - integral e^(2x) d x \
+  &= (x+2) dot (e^(2x) dot 1/2x) - e^(2x) dot 1/2x \
+  &= x dot (e^(2x) dot 1/2x) + e^(2x) dot 1/2x \
+$