Question

`int"x"^"n"."log"  "x"  "dx"`

Answer 1

`int"x"^"n"."log"  "x"  "dx"`

= `int"log" "x"  "x"^"n"  "dx"`

`int"u"."v" "dx" = "u" int "v" "dx" - int ("du"/"dx") [int "v dx"] "dx"`

= `"log x" int "x"^"n" "dx" - int ["d"/"dx" ("log x")int "x"^"n" "dx"] "dx"`

= `"log x" xx ("x"^("n" + 1))/("n" + 1) - int 1/"x".("x"^("n" + 1))/("n" + 1) "dx"`

= `("x"^("n" + 1) "log x")/("n" + 1) - 1/("n" + 1) int"x"^"n" "dx"`

= `("x"^("n" + 1) "log x")/("n" + 1) - ("x"^("n" + 1))/("n" + 1)^2 + "C"`

= `("x"^("n" + 1) "log x")/("n" + 1) ["log x" - 1/("n" + 1)] + "C"`