質問#f9641

質問#f9641
Anonim

回答:

#int cos(x)/(sin ^ 2(x)+ sin(x)) "d" x = ln | sin(x)/(sin(x)+1)| + C#

説明:

# int cos(x)/(sin ^ 2(x)+ sin(x)) "d" x#

代替 #u = sin(x)# そして # "d" u = cos(x) "d" x#。これは与える

#= int ( "d" u)/(u ^ 2 + u)#

#= int ( "d" u)/(u(u + 1))#

部分分数に分離する #1 /(u(u + 1))= 1 / u-1 /(u + 1)#:

#= int (1 / u-1 /(u + 1)) "d" u#

#= ln | u | -ln | u + 1 | + C#

#= ln | u /(u + 1)| + C#

代用 #u = sin(x)#:

#= ln | sin(x)/(sin(x)+1)| + C#