Integrate: `int cos^3 x sin^4 x dx.`

Answer

To solve the integral \( \int \cos^3 x \sin^4 x \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We can express the integral as:
\[ \int \cos^3 x \sin^4 x \, dx = \int \sin^4 x \cos^2 x \cos x \, dx \]
Here, we have separated \( \cos^3 x \) into \( \cos^2 x \cdot \cos x \).

### Step 2: Use the Pythagorean Identity
Using the identity \( \cos^2 x = 1 - \sin^2 x \), we can rewrite the integral:
\[ \int \sin^4 x (1 - \sin^2 x) \cos x \, dx \]

### Step 3: Substitute \( t = \sin x \)
Let \( t = \sin x \). Then, \( dt = \cos x \, dx \). The integral becomes:
\[ \int t^4 (1 - t^2) \, dt \]

### Step 4: Expand the Integral
Now, we expand the integrand:
\[ \int (t^4 - t^6) \, dt \]

### Step 5: Integrate Term by Term
Now we can integrate term by term:
\[ \int t^4 \, dt - \int t^6 \, dt = \frac{t^5}{5} - \frac{t^7}{7} + C \]

### Step 6: Combine the Results
Combining the results, we have:
\[ \frac{t^5}{5} - \frac{t^7}{7} + C \]

### Step 7: Substitute Back for \( t \)
Now we substitute back \( t = \sin x \):
\[ \frac{\sin^5 x}{5} - \frac{\sin^7 x}{7} + C \]

### Step 8: Simplify the Expression
To combine the terms, we can express it as:
\[ \frac{1}{35} (7 \sin^5 x - 5 \sin^7 x) + C \]

Thus, the final answer is:
\[ \int \cos^3 x \sin^4 x \, dx = \frac{1}{35} (7 \sin^5 x - 5 \sin^7 x) + C \]