How do you test the improper integral #int (2x-1)^-3dx# from #[0,1/2]# and evaluate if possible?

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Answer 1 · 2018-02-19T07:03:32

The integral is divergent.

To calculate the improper integral, proceed as follows :

First, calculate the indefinite integral

#int_0^t(2x-1)^-3dx#

#=[-1/4(2x-1)^-2]_0^t#

#=[1/4(2x-1)^-2]_t^0#

#=1/4-1/4(2t-1)^-2#

Second, determine the limits as #t->1/2#

#lim_(t->1/2)int_0^t(2x-1)^-3dx#

#=lim_(t->1/2)(1/4-1/4(2t-1)^-2)#

#=1/4-1/4(1/0)^2#

#=-oo#