Question

test the convergence of integral sign of integration limit 1 to infinity log x/ x^2 dx​

Answer 1

Answer:

I tried to solve it in an intuitive manner, but I am not sure if it's right or wrong. Some feedback would be lovely!

This is how I approached the problem.

Step 1: I used integration by parts.

∫∞1ln(x)x2dx=∫∞1ln(x)ddx(−1x)dx=−ln(x)x∣∣∞1+∫∞11x2dx

Step 2: Verify if ∫∞11x2dx converges or not.

Fact:

∫∞11xpdx for p > 1 the area under the graph is finite and the integral converges.

In our case we have: ∫∞11x2dx where 2 > 1 ⟹Fact ∫∞11x2dx converges.

Step 3: Let's see what happens with −ln(x)x∣∣∞1

If we take

limb→∞−ln(x)x∣∣b1⟹−limb→∞ln(b)b−0⟹L′Hopital−limb→∞1b=0

Therefore I concluded that this part: −ln(x)x∣∣∞1 does not affect my convergence since it's zero.

Finally from steps (1), (2) and (3) we can see that ∫∞1ln(x)x2dx converges.