The integral to solve is:

$\int_0^{\infty} 7x \, dx$

To evaluate this, let's break it down step by step:

1. Factor out the constant $7$:

$7 \int_0^{\infty} x \, dx$

2. Now, calculate the integral of $x$. The indefinite integral of $x$ is:

$\frac{x^2}{2}$

3. Applying the limits of integration from $0$ to $\infty$:

$7 \left[ \frac{x^2}{2} \right]_0^{\infty}$

4. Evaluating at the upper and lower limits:

• At $x = \infty$, $\frac{x^2}{2} \to \infty$.
• At $x = 0$, $\frac{x^2}{2} = 0$.

Thus, we have:

$7 \left[ \frac{\infty^2}{2} - \frac{0^2}{2} \right] = 7 \times \infty$

Since this expression diverges to infinity, the integral does not converge. Therefore, the result is:

$\text{The integral diverges to } \infty.$

Would you like further clarification or more details on this?

Follow-up questions:

1. What is the significance of divergence in integrals?
2. How do we know if an integral converges or diverges?
3. Can we apply any conditions to make this integral convergent?
4. What is the difference between proper and improper integrals?
5. How does changing the limits of integration affect the outcome?

Tip:

When evaluating integrals with infinity as a limit, it's crucial to check for convergence by analyzing the behavior of the function at the boundaries.