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

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Answer 1 · 2018-02-21T07:06:18

The integral diverges

To calculate the improper integral, proceed as follows

#int_(-1)^1 2x^-3dx#

#=lim_(t->O^-)(int_-1^t 2x^-3dx)+lim_(t->O^+)(int_t^1 2x^-3dx)#

The first part is

#lim_(t->O^-)(int_-1^t2x^-3dx)#

#=lim_(t->O^-) [-1/x^2]_1^t#

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

#=+oo#

The integral diverges

The second part is

#lim_(t->O^+)(int_t^1x^-3dx)#

#=lim_(t->O^+) [-1/x^2]_t^1#

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

#=+oo#

The integral diverges