Let's analyze and solve the integral given in the question:

$\int_{0}^{\pi/2} \frac{\cos(x)}{(\sin(x) + 3)^2} dx$

This integral is in a standard form suitable for substitution. Let's proceed step by step.

Step 1: Substitution

Let: $u = \sin(x) + 3 \implies \frac{du}{dx} = \cos(x) \quad \text{or} \quad du = \cos(x)dx$

When $x = 0$, $u = \sin(0) + 3 = 3$.

When $x = \frac{\pi}{2}$, $u = \sin\left(\frac{\pi}{2}\right) + 3 = 4$.

Thus, the integral becomes: $\int_{3}^{4} \frac{1}{u^2} du$

Step 2: Simplify the integral

The integral of $u^{-2}$ is: $\int u^{-2} du = -\frac{1}{u}$

Using this result, the integral becomes: $-\frac{1}{u} \Big|_{3}^{4}$

Step 3: Evaluate the definite integral

Substituting the limits: $-\frac{1}{4} + \frac{1}{3} = \frac{1}{3} - \frac{1}{4} = \frac{4}{12} - \frac{3}{12} = \frac{1}{12}$

Final Answer:

The value of the integral is: $\boxed{\frac{1}{12} \text{ (Option c)}}.$

Let me know if you'd like a further breakdown or have any questions!

Follow-Up Questions:

1. How does substitution simplify definite integrals like this?
2. What is the role of adjusting the limits in substitution?
3. Why is the antiderivative of $u^{-2}$ equal to $-\frac{1}{u}$?
4. Can this integral be solved without substitution? How?
5. What other techniques might be helpful in evaluating definite integrals?

Tip:

Always check your limits carefully when performing a substitution in definite integrals—they ensure your solution stays consistent!