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Compute the value of the following improper integral if it converges. If it diverges, enter INF if it diverges
Compute the value of the following improper integral if it converges. If it diverges, enter INF if it diverges to infinity, MINF if it diverges to minus infinity, or DIV otherwise (hint: integrate by parts).
integral(4 ln(x)/x^3, x, 1, infinity)
lim[x->inf] 4ln(x)/x^3=0 ==>
the integral could converge
Use partial integration:
∫u dv=uv-∫v du
∫[1;inf] 4ln(x)/x^3 dx=
lim[a->inf] ∫[1;a] 4ln(x)/x^3 dx=
lim[a->inf] ∫[1;a] 2ln(x) d(-1/x^2)=
lim[a->inf] {-2ln(x)/x^2 [1;a] +2∫[1;a] 1/x^3 dx}=
lim[a->inf] {-2ln(a)/a^2 -2/(2x^2) [1;a]}=
0 - lim[a->inf] 1/(a^2) +1= 1
If you need explanations, ask.
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