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I'm having trouble with this integral:

integral of (1/x^2)*sec(1/x)*tan(1/x)dx

i let u=(1/x)

so du=(1/x^2)dx

so du(x^2)=dx

If I do this, the integral works out perfectly since the x^2 cancel out

so it equals sec(u) + C

HOWEVER, my question is

if i let u = (1/x)

why don't i replace the (1/x^2) (in the beginning of the integral) with u^2

CAN SOMEONE HELP ME?!?!