To integrate \( \sin^5(x) \cdot \cos^4(x) \, dx \), we can use the following approach:

### Step 1: Express the sine and cosine powers in terms of sine and cosine functions
We will separate one \(\sin(x)\) and one \(\cos(x)\) from the original integral to make the substitution easier:

\[
\sin^5(x) \cdot \cos^4(x) = \sin^4(x) \cdot \sin(x) \cdot \cos^4(x)
\]

This can be written as:

\[
\sin^4(x) \cdot \cos^4(x) \cdot \sin(x) \, dx
\]

### Step 2: Use trigonometric identities
We can rewrite \(\sin^4(x)\) as \((\sin^2(x))^2\) and \(\cos^4(x)\) as \((\cos^2(x))^2\):

\[
(\sin^2(x))^2 \cdot (\cos^2(x))^2 \cdot \sin(x) \, dx
\]

Now, recall that \(\sin^2(x) = 1 - \cos^2(x)\). Let's make the substitution \(u = \cos(x)\), so that \(du = -\sin(x) \, dx\). The integral now becomes:

\[
\int (1 - u^2)^2 \cdot u^4 \cdot (-du)
\]

### Step 3: Simplify and expand the expression
Expanding \((1 - u^2)^2\) gives:

\[
(1 - u^2)^2 = 1 - 2u^2 + u^4
\]

So the integral becomes:

\[
-\int (1 - 2u^2 + u^4) \cdot u^4 \, du
\]

Expanding this:

\[
-\int \left( u^4 - 2u^6 + u^8 \right) \, du
\]

### Step 4: Integrate term by term
Now integrate each term separately:

\[
-\left[ \frac{u^5}{5} - \frac{2u^7}{7} + \frac{u^9}{9} \right] + C
\]

Finally, substitute back \(u = \cos(x)\):

\[
-\left[ \frac{\cos^5(x)}

Explore how to integrate sin^5(x) * cos^4(x) using trigonometric identities and the substitution method. This problem highlights advanced trigonometric integrals and is suitable for students in Grades 11-12.

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