is the integral convergent or divergent?

Question

You are going to have to forgive me as I dont know how to add the Integral symbol so I will use S

S((1+x)/(x^2+2x))dx  for 1 to infinity

my solution
1)  S(x)/(x^2+2x)dx +  S1/(x^2+2x)dx

Integrate both terms

ln(x+2) + ln(x^2+2x)
then I take the limit from 1 to infinity for both terms and add.
this is my answer it is not the same answer as the book but is it wrong??

The book answer
S((1+x)/(x^2+2x))dx = 1/2S((1+x)/(x^2+2x))dx
integrate to get
1/2ln(x^2+2x)

Both answers are divergent

kb · Accepted Answer

Your integrals are not right at all.

However, we can show that this integral diverges by the Comparison Test.
(1+x)/(x^2+2x) > x/(x^2 + 2x) > x/(x^2 + 2x^2) = (1/3) * 1/x for all x > 1.

Since ∫(1 to ∞) (1/3) dx/x = (1/3) ln |x| {for x = 1 to ∞} = ∞ (divergent),
the original integral must also diverge.

I hope this helps!

is the integral convergent or divergent?

1 Answer