`inte^(4-3x) dx `

Answer

To solve the integral \( \int e^{4 - 3x} \, dx \), we can follow these steps:

### Step 1: Identify the function to integrate
We have the function \( e^{4 - 3x} \).

### Step 2: Use substitution
Let \( u = 4 - 3x \). Then, we need to find \( du \):
\[ du = -3 \, dx \quad \Rightarrow \quad dx = -\frac{1}{3} \, du \]

### Step 3: Substitute in the integral
Now, substituting \( u \) and \( dx \) into the integral:
\[ \int e^{4 - 3x} \, dx = \int e^u \left(-\frac{1}{3} \, du\right) = -\frac{1}{3} \int e^u \, du \]

### Step 4: Integrate \( e^u \)
The integral of \( e^u \) is simply \( e^u \):
\[ -\frac{1}{3} \int e^u \, du = -\frac{1}{3} e^u + C \]

### Step 5: Substitute back to original variable
Now, substitute back \( u = 4 - 3x \):
\[ -\frac{1}{3} e^{4 - 3x} + C \]

### Final Answer
Thus, the integral \( \int e^{4 - 3x} \, dx \) is:
\[ -\frac{1}{3} e^{4 - 3x} + C \]

---