$$ 56\int_0^{\infty}\!55x\,\mathrm{d}x $$

The given integral is:

\[
56 \int_0^{\infty} 55x \, \mathrm{d}x
\]

First, we can factor out the constants from the integral:

\[
56 \times 55 \int_0^{\infty} x \, \mathrm{d}x
\]

Simplifying the constants:

\[
3080 \int_0^{\infty} x \, \mathrm{d}x
\]

Now, let's evaluate the integral $ \int_0^{\infty} x \, \mathrm{d}x $. This is a divergent integral because as $x \to \infty$, $ x $ grows without bound. Therefore, the integral does not converge to a finite value and the result is:

\[
\boxed{\infty}
\]

This means that the original integral is divergent.

Would you like more details on improper integrals or any related concepts?

Here are 5 related questions to explore:
1. What are improper integrals, and how do they differ from definite integrals?
2. How do we determine when an integral is divergent?
3. What methods can be used to evaluate improper integrals that do converge?
4. Can divergent integrals be assigned a value using special techniques (e.g., regularization)?
5. How do integrals behave when the integrand grows at a rate proportional to $ x $?

**Tip:** When evaluating integrals that involve infinity as a limit, always check whether the function converges or diverges before proceeding.

Explore the improper integral 56∫₀^∞ 55x dx. This step-by-step guide covers the concept of improper integrals and explains why the integral diverges. Learn about divergence and how constants factor into integral evaluation. Suitable for undergraduate-level calculus students.

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