How do I determine whether $\int _{ -\infty }^{ \infty }{ \frac { x }{ 37+x^2 } dx } $ is convergent or divergent?

Question

Determine whether the integral is convergent or divergent.

$$\int _{ -\infty }^{ \infty }{ \frac { x }{ 37+x^2 } dx } $$

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What I did:

Let $u=37+x^2$, $du=2xdx$

So, we substitute and split the integral to get:

$$\frac { 1 }{ 2 } \int _{ -\infty }^{ \infty }{ \frac { 1 }{ u } du } =\frac { 1 }{ 2 } \int _{ -\infty }^{ 0 }{ \frac { 1 }{ u } du } +\frac { 1 }{ 2 } \int _{ 0 }^{ \infty }{ \frac { 1 }{ u } du } $$

$$\lim _{ t\rightarrow -\infty }{ \frac { 1 }{ 2 } \int _{ t }^{ 0 }{ \frac { 1 }{ u } du } } +\lim _{ t\rightarrow \infty }{ \frac { 1 }{ 2 } \int _{ 0 }^{ t }{ \frac { 1 }{ u } du } } $$

$$\lim _{ t\rightarrow -\infty }{ [\frac { 1 }{ 2 } \ln { (37+x^{ 2 }) } ] } _{ t }^{ 0 }+\lim _{ m\rightarrow \infty }{ [\frac { 1 }{ 2 } \ln { (37+x^{ 2 }) } ] } _{ 0 }^{ m }$$

$$\frac { 1 }{ 2 } \lim _{ t\rightarrow -\infty }{ [\ln { (37) } -\ln { (37+t^{ 2 }) } ] } +\frac { 1 }{ 2 } \lim _{ m\rightarrow \infty }{ [\ln { (37+m^{ 2 })-\ln { (37) } } ] } $$

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This is as far as I have come. I did attempt to evaluate what will happen at $-\infty$ and $\infty$ by plugging in those values, but the end result did not make sense to me. I would like to know what I need to do to either complete what I have already done, or what I should do if what i have done thus far is wrong.

Answer 1

As you noted $$\int_{-t}^{m}\frac{x}{37+x^2} dx=\frac{1}{2}\ln(37+m^{2})-\frac{1}{2}\ln(37+t^{2})=\frac{1}{2}\ln\left(\frac{37+m^{2}}{37+t^{2}}\right).$$ What happens when $m=t\to +\infty$? What happens when $m=2t\to +\infty$?

Notice that if the integral is convergent, those limits should be equal. See here for details.

Answer 2

The problem is that your substitutions is not a bijective function, so you can't really solve your problem like that. You would have to also replace the bounds of the integration and get a nonsensical expression $\int_\infty^\infty\dots dx$.

To calculate it, you would have to split the integral into two parts on which $37+x^2$ is bijective, so

$$\int_{-\infty}^0 \dots dx + \int_0^\infty\dots dx$$

However, there are also methods you can use to determine the convergence that do not require you to actually calculate the integral.

For example, you may be familiar with the theorem that if $f$ is bounded below by some function which takes the form $$\frac{g(x)}{x^s}$$ where $g(x)>m>0$ for all $x>0$, then the integral $$\int_0^\infty f(x)dx$$ converges if $s>1$ and diverges if $s\leq 1$.