Let's solve the given problems step by step.

Problem 8(a):

$\int \frac{49x}{\sqrt{1 - 49x^2}} \, dx$

Solution:

1. Let $u = 49x$, so $du = 49 \, dx$. Then, the integral becomes: $\int \frac{u}{\sqrt{1 - u^2}} \, \frac{du}{49}.$

2. Factor out $\frac{1}{49}$: $\frac{1}{49} \int \frac{u}{\sqrt{1 - u^2}} \, du.$

3. The antiderivative of $\frac{u}{\sqrt{1 - u^2}}$ is $-\sqrt{1 - u^2} + C$.

4. Substitute back $u = 49x$: $-\sqrt{1 - (49x)^2} + C.$

Final answer: $\boxed{-\sqrt{1 - 49x^2} + C \quad \text{(Option D)}.}$

Problem 8(b):

$\int \frac{25}{\sqrt{1 - 25x^2}} \, dx$

Solution:

1. This is a standard arcsine integral form: $\int \frac{a}{\sqrt{1 - (ax)^2}} \, dx = \arcsin(ax) + C.$

2. Here $a = 5$. Applying the formula: $5 \arcsin(5x) + C.$

Final answer: $\boxed{5\arcsin(5x) + C \quad \text{(Option G)}.}$

Would you like further clarifications or step-by-step explanations? Here are 5 related questions to expand your understanding:

1. How does the substitution method work in integration?
2. Why is the arcsine function important in integral calculus?
3. Can you derive the general formula for integrals of the form $\int \frac{a}{\sqrt{1 - (ax)^2}} \, dx$?
4. What are the common pitfalls in solving integrals involving square roots?
5. How can you verify your answer for such integrals?

Tip: For any integral involving $\sqrt{1 - x^2}$, always consider trigonometric identities or substitutions like arcsine or cosine for simplification.