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The given integral is:

\[
\int \frac{\sin(\sqrt{\theta})}{\sqrt{\theta} \cos^3(\sqrt{\theta})} d\theta.
\]

### Solution:

We can perform a substitution to simplify this expression.

Let:
\[
u = \sqrt{\theta} \implies \theta = u^2, \quad \frac{d\theta}{du} = 2u \implies d\theta = 2u du.
\]

Substitute \( \theta = u^2 \) and \( d\theta = 2u du \) into the integral:

\[
\int \frac{\sin(u)}{u \cos^3(u)} \cdot 2u \, du = 2 \int \frac{\sin(u)}{\cos^3(u)} \, du.
\]

Now we need to solve \( 2 \int \frac{\sin(u)}{\cos^3(u)} \, du \).

This integral can be tackled by using a trigonometric identity. Recall:
\[
\frac{\sin(u)}{\cos^3(u)} = \frac{1}{\cos^2(u)} \cdot \frac{\sin(u)}{\cos(u)} = \sec^2(u) \tan(u).
\]

Thus, the integral becomes:
\[
2 \int \sec^2(u) \tan(u) \, du.
\]

Now we recognize that the derivative of \( \tan(u) \) is \( \sec^2(u) \), so:

\[
2 \int \sec^2(u) \tan(u) \, du = 2 \cdot \frac{1}{2} \sec^2(u) + C = \sec^2(u) + C.
\]

Finally, substitute back \( u = \sqrt{\theta} \) to get the final answer:

\[
\sec^2(\sqrt{\theta}) + C.
\]

### Final Answer:
\[
\sec^2(\sqrt{\theta}) + C.
\]

Let me know if you want further details or have any questions.

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Here are five related questions:
1. How does substitution help in solving integrals involving radicals?
2. What is the derivative of \( \tan(\theta) \) and how does it simplify integrals involving trigonometric functions?
3. Can you use integration by parts for this integral? Why or why not?
4. What are other methods for solving trigonometric integrals?
5. How would you solve an integral of the form \( \int \frac{\sin^2(x)}{\cos^4(x)} \, dx \)?

**Tip**: Always check for possible substitutions or identities when dealing with trigonometric integrals.

∫ (sin(√θ) / (√θ cos^3(√θ))) dθ

This problem demonstrates the integration of sin(√θ) / (√θ cos^3(√θ)) by using the substitution u = √θ. The solution involves trigonometric identities and substitution techniques, ideal for students studying advanced calculus.

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