27.

1848 days 전, rksekwnwls@gmail.com 작성

int _{} ^{} {{x ^{3} +x ^{2} +2x+1} over {(x ^{2} +1)(x ^{2} +2)} dx}# # {x ^{3} +x ^{2} +2x+1} over {(x ^{2} +1)(x ^{2} +2)}# = {ax+b} over {(x ^{2} +1)} + {cx+d} over {(x ^{2} +2)}# = {(ax+b)(x ^{2} +2)+(cx+d)(x ^{2} +1)} over {(x ^{2} +1)(x ^{2} +2)}# = {ax ^{3} +bx ^{2} +2ax+2b+cx ^{3} +dx ^{2} +cx+d} over {(x ^{2} +1)(x ^{2} +2)}# = {(a+c)x ^{3} +(b+d)x ^{2} +(2a+c)x+2b+d} over {(x ^{2} +1)(x ^{2} +2)}# # a+c=1 CDOTS ➀# b+d=1 CDOTS ➁# 2a+c=2 CDOTS ➂# 2b+d=1` CDOTS `➃# # ➂-➀# a=1` CDOTS `➄# c=0`( BECAUSE `➀,➄)` CDOTS `➅# # ➃-➁# b=0` CDOTS `➆`# d=1( BECAUSE `➁,➆)` CDOTS `➇# # {x ^{3} +x ^{2} +2x+1} over {(x ^{2} +1)(x ^{2} +2)}# = {x} over {(x ^{2} +1)} + {1} over {(x ^{2} +2)}# # int _{} ^{} {{{x ^{3} +x ^{2} +2x+1} over {(x ^{2} +1)(x ^{2} +2)}}dx} = int _{} ^{} {{{x} over {x^2+1}} + {1} over {x^{2} +2}}dx} = {1} over {2} ln LEFT | x^2+1 RIGHT| + {{sqrt{2}} over {2}}arctan({{sqrt{2}} over {{2}}x) + C