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The given integral is:

\[
\int_0^{\pi/2} \sin^5(\theta) \cos^5(\theta) \, d\theta
\]

To solve this, we can apply a common trigonometric identity and a substitution. First, recognize the product of powers of sine and cosine can be simplified using the identity:

\[
\sin(\theta) \cos(\theta) = \frac{1}{2} \sin(2\theta)
\]

Thus, we rewrite the integrand:

\[
\sin^5(\theta) \cos^5(\theta) = \left(\sin(\theta) \cos(\theta)\right)^5 = \left(\frac{1}{2} \sin(2\theta)\right)^5 = \frac{1}{32} \sin^5(2\theta)
\]

Now, the integral becomes:

\[
\int_0^{\pi/2} \frac{1}{32} \sin^5(2\theta) \, d\theta
\]

Next, we use a reduction formula for integrals of the form \(\int \sin^n(x) \, dx\). The reduction formula for \(\sin^n(x)\) is:

\[
\int \sin^n(x) \, dx = -\frac{1}{n} \sin^{n-1}(x) \cos(x) + \frac{n-1}{n} \int \sin^{n-2}(x) \, dx
\]

Let's now proceed to solve this integral step-by-step.The value of the integral is:

\[
\int_0^{\pi/2} \sin^5(\theta) \cos^5(\theta) \, d\theta = \frac{1}{60} \approx 0.01667
\]

Would you like any further details on the steps, or do you have any questions?

Here are some related questions you might find useful:

1. How does the reduction formula work for integrals of sine and cosine?
2. What are other methods for solving integrals involving powers of trigonometric functions?
3. Can you explain how to solve similar integrals using substitution?
4. How does the identity \(\sin(\theta)\cos(\theta) = \frac{1}{2}\sin(2\theta)\) help simplify trigonometric integrals?
5. What are some common applications of integrals like these in physics or engineering?

**Tip:** When working with powers of sine and cosine, converting the product into terms of \(\sin(2\theta)\) or using double-angle identities can greatly simplify the integration.

Evaluate the integral \( \int_0^{\pi/2} \sin^5(\theta) \cos^5(\theta) \, d\theta \).

Explore a detailed solution for evaluating the integral \( \int_0^{\pi/2} \sin^5(\theta) \cos^5(\theta) \, d\theta \) using trigonometric identities and reduction formulas. Understand the process of simplifying the integrand and applying the reduction formula for powers of sine functions.

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