# How do you test the improper integral #int (3x)/(x+1)^4 dx# from #[0, oo)# and evaluate if possible?

First just treating without the bounds:

Rewriting and integrating:

So, we have:

In order to "evaluate" this at infinity, take the limit:

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To test the improper integral ∫(3x)/(x+1)^4 dx from [0, ∞):

- Check if the integral is improper due to infinity in the interval.
- Determine if the integrand has singularities in the interval.
- Apply the limit definition of improper integrals to evaluate the integral.

To evaluate the integral if possible:

- Use partial fraction decomposition to rewrite the integrand.
- Integrate each term separately.
- Apply limits as needed.

The solution involves some algebraic manipulation, and the final result may be expressed in terms of natural logarithms and limits.

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