It seems like you are referring to an integral of the form:

\[
\int_{\frac{\pi}{2}}^{0} \cos(7x) \sin(5x) \, dx
\]

Let's solve this step by step.

### Step 1: Use a trigonometric identity
The product of \(\cos(7x)\) and \(\sin(5x)\) can be simplified using a product-to-sum identity:

\[
\cos(A) \sin(B) = \frac{1}{2} [\sin(A + B) - \sin(A - B)]
\]

Here, \(A = 7x\) and \(B = 5x\), so:

\[
\cos(7x) \sin(5x) = \frac{1}{2} [\sin(12x) - \sin(2x)]
\]

### Step 2: Rewrite the integral
Now, substitute this identity into the integral:

\[
\int_{\frac{\pi}{2}}^{0} \cos(7x) \sin(5x) \, dx = \frac{1}{2} \int_{\frac{\pi}{2}}^{0} [\sin(12x) - \sin(2x)] \, dx
\]

### Step 3: Integrate each term
Now, integrate each term separately.

For \(\sin(12x)\):

\[
\int \sin(12x) \, dx = -\frac{1}{12} \cos(12x)
\]

For \(\sin(2x)\):

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

### Step 4: Substitute the limits of integration
Now apply the limits \(\frac{\pi}{2}\) to 0 to both integrals.

For \(\sin(12x)\):

\[
\left[-\frac{1}{12} \cos(12x) \right]_{\frac{\pi}{2}}^0 = -\frac{1}{12} [\cos(0) - \cos(6\pi)] = -\frac{1}{12} [1 - 1] = 0
\]

For \(\sin(2x)\):

\[
\left[-\frac{1}{2} \cos(2x) \right]_{\frac{\pi}{2}}^0 = -\frac{1}{2} [\cos(0) - \cos(\pi)] = -\frac{1}{2} [1 - (-1)] = -\frac{1}{2} \times 2 = -1
\]

### Step 5: Combine the results
Now, combine the two results:

\[
\frac{1}{2} [0 - (-1)] = \frac{1}{2} \times 1 = \frac{1}{2}
\]

### Final Answer:
The value of the integral is \(\frac{1}{2}\).

Do you have any questions or want further details?

Here are five related questions:
1. How would the result change if the limits of integration were reversed?
2. What happens if we switch the coefficients in the trigonometric functions?
3. Can this be solved using complex exponentials?
4. How can integration by parts be applied to this problem?
5. What are the general applications of product-to-sum identities in integrals?

**Tip:** Always check for trigonometric identities when dealing with products of sines and cosines in integrals!

This solution explains how to solve the integral ∫π/20 cos(7x) sin(5x) dx step by step, using trigonometric identities. Learn how to simplify the product of sine and cosine using the product-to-sum identity, and then integrate the result. Suitable for undergraduate calculus students.

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