`int(3-5x)^(7)dx=?`

Options

A. `-5(3-5x)^(6)+C`
B. `((3-5x)^(8))/(-40)+C`
C. `(-5(3-5x)^(8))/(8)+C`
D. none of these

Explanation

To solve the integral \( \int (3 - 5x)^7 \, dx \), we will use the substitution method. Here are the steps:

### Step-by-Step Solution:

1. **Substitution**: Let \( t = 3 - 5x \).
- Then, differentiate \( t \) with respect to \( x \):
\[ \frac{dt}{dx} = -5 \implies dt = -5 \, dx \implies dx = -\frac{dt}{5} \]

2. **Rewrite the Integral**: Substitute \( t \) and \( dx \) into the integral:
\[ \int (3 - 5x)^7 \, dx = \int t^7 \left(-\frac{dt}{5}\right) \]
This simplifies to:
\[ -\frac{1}{5} \int t^7 \, dt \]

3. **Integrate**: Now integrate \( t^7 \):
\[ -\frac{1}{5} \cdot \frac{t^8}{8} + C = -\frac{1}{40} t^8 + C \]

4. **Back Substitute**: Replace \( t \) back with \( 3 - 5x \):
\[ -\frac{1}{40} (3 - 5x)^8 + C \]

### Final Answer:
Thus, the integral \( \int (3 - 5x)^7 \, dx \) is:
\[ -\frac{1}{40} (3 - 5x)^8 + C \]