Question #0228a

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Answer 1 · 2016-08-14T19:18:33

#(1) : int(1-x)^ndx=-(1-x)^(n+1)/(n+1)+C, if, n!=-1#

#(2) : int(1-x)^ndx=-ln|1-x|+C', or, ln|1/(1-x)|+C', if n=-1#.

Let #I=int(1-x)^ndx#

Take, #(1-x)=t rArr -dx=dt#

Hence, #I=int t^n(-dt)=-int t^ndt#

#=-t^(n+1)/(n+1), if, n+1!=0, i.e., n!=-1#

#=-(1-x)^(n+1)/(n+1)+C, if, n!=-1#

In case, #n=-1,

#I=intt^-1(-dt)=-int1/tdt=-ln|t|=-ln|1-x|+C'#,

or, #=ln|1/(1-x)|+C'#