If `intsin^(5)x cos^(4)x dx =A cos^(9) x +B cos^(7)x+C cos^(5)x+D`,then 9A +7B +5C=

Options

A. 1
B. 0
C. `-1`
D. none of these

Explanation

To solve the integral \( \int \sin^5 x \cos^4 x \, dx \) and express it in the form \( A \cos^9 x + B \cos^7 x + C \cos^5 x + D \), we will follow these steps:

### Step 1: Rewrite the integral
We start with the integral:
\[ \int \sin^5 x \cos^4 x \, dx \]
We can express \( \sin^5 x \) as \( \sin^4 x \sin x \) and rewrite \( \sin^4 x \) using the identity \( \sin^2 x = 1 - \cos^2 x \):
\[ \sin^4 x = (1 - \cos^2 x)^2 = 1 - 2\cos^2 x + \cos^4 x \]
Thus, we have:
\[ \sin^5 x = (1 - 2\cos^2 x + \cos^4 x) \sin x \]
So, the integral becomes:
\[ \int (1 - 2\cos^2 x + \cos^4 x) \sin x \cos^4 x \, dx \]

### Step 2: Substitute \( u = \cos x \)
Let \( u = \cos x \), then \( du = -\sin x \, dx \) or \( \sin x \, dx = -du \). The integral now transforms to:
\[ -\int (1 - 2u^2 + u^4) u^4 \, du \]
This simplifies to:
\[ -\int (u^4 - 2u^6 + u^8) \, du \]

### Step 3: Integrate term by term
Now we can integrate each term:
\[ -\left( \frac{u^5}{5} - \frac{2u^7}{7} + \frac{u^9}{9} \right) + C \]
This results in:
\[ -\frac{u^5}{5} + \frac{2u^7}{7} - \frac{u^9}{9} + C \]

### Step 4: Substitute back \( u = \cos x \)
Substituting back \( u = \cos x \):
\[ -\frac{\cos^5 x}{5} + \frac{2\cos^7 x}{7} - \frac{\cos^9 x}{9} + C \]

### Step 5: Rearranging the expression
We can rearrange the expression to match the form \( A \cos^9 x + B \cos^7 x + C \cos^5 x + D \):
\[ -\frac{1}{9} \cos^9 x + \frac{2}{7} \cos^7 x - \frac{1}{5} \cos^5 x + C \]
From this, we identify:
- \( A = -\frac{1}{9} \)
- \( B = \frac{2}{7} \)
- \( C = -\frac{1}{5} \)

### Step 6: Calculate \( 9A + 7B + 5C \)
Now we compute:
\[ 9A + 7B + 5C = 9\left(-\frac{1}{9}\right) + 7\left(\frac{2}{7}\right) + 5\left(-\frac{1}{5}\right) \]
Calculating each term:
- \( 9A = -1 \)
- \( 7B = 2 \)
- \( 5C = -1 \)

Adding these together:
\[ -1 + 2 - 1 = 0 \]

### Final Answer
Thus, the value of \( 9A + 7B + 5C \) is:
\[ \boxed{0} \]