I am trying to find whether the integral from 0 to infinity of ln(x^2)/e^(x^2) is convergent or divergent.
Firstly, I broke the integral up into two improper integrals, namely:
I am trying to use the Comparison Test, and was notified by someone that the following worked:
> on (0,1) the exact value of ln(x) < 1/x^(1/2) AND exact value of e^(-x^2) is less than or equal to 1
> on (1,inf), exact value of ln(x) < x < 2*e^(x/2) and e^(-x^2)<e^(-x)
I am confused on why they are taking the exact value and how they could imply the above inequalities.
However, I do know how to prove for (1,inf) that e^(-x^2)<e^(-x)
x=1
=> x^2>x
=> e^(x2)>e^(x)
=> 1/e^(x) > 1/e^(x^2)
but the rest I am so unsure on.
Can someone please help me prove this and explain the reason for the exect values?
Firstly, I broke the integral up into two improper integrals, namely:
- the integral from 0 to 1;
- the integral from 1 to infinity
I am trying to use the Comparison Test, and was notified by someone that the following worked:
> on (0,1) the exact value of ln(x) < 1/x^(1/2) AND exact value of e^(-x^2) is less than or equal to 1
> on (1,inf), exact value of ln(x) < x < 2*e^(x/2) and e^(-x^2)<e^(-x)
I am confused on why they are taking the exact value and how they could imply the above inequalities.
However, I do know how to prove for (1,inf) that e^(-x^2)<e^(-x)
x=1
=> x^2>x
=> e^(x2)>e^(x)
=> 1/e^(x) > 1/e^(x^2)
but the rest I am so unsure on.
Can someone please help me prove this and explain the reason for the exect values?