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

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Answer 1 · 2017-05-16T18:19:59

Integrate by substitution and evalute.

#int_-oo^0 (2x-1)^-3dx = lim_(brarr-oo)int_b^0 (2x-1)^-3dx#

# = lim_(brarr-oo) 1/2 int_b^0 (2x-1)^-3 2dx#

Let #u = 2x-1# so that #du = 2u# and #int u^-3 du = -1/2u^-2+C#

So,

#lim_(brarr-oo) 1/2 int_b^0 (2x-1)^-3 2dx = lim_(brarr-oo)[-1/4(2x-1)^-2]_b^0#

# = lim_(brarr-oo)(-1/4 +1/(4(2b-1)^2))#

# = -1/4#