Integrate the following: `int(2t-4)^(-4)dt`

Explanation

To solve the integral \( \int (2t - 4)^{-4} dt \), we will follow these steps:

### Step 1: Substitution
Let us make a substitution to simplify the integral. We can let:
\[ z = 2t - 4 \]
Then, we differentiate both sides to find \( dt \):
\[ dz = 2 dt \quad \Rightarrow \quad dt = \frac{dz}{2} \]

### Step 2: Rewrite the Integral
Now we can rewrite the integral in terms of \( z \):
\[ \int (2t - 4)^{-4} dt = \int z^{-4} \cdot \frac{dz}{2} \]
This simplifies to:
\[ \frac{1}{2} \int z^{-4} dz \]

### Step 3: Integrate
Next, we integrate \( z^{-4} \):
\[ \int z^{-4} dz = \frac{z^{-3}}{-3} + C = -\frac{1}{3} z^{-3} + C \]
Thus, we have:
\[ \frac{1}{2} \left(-\frac{1}{3} z^{-3} + C\right) = -\frac{1}{6} z^{-3} + \frac{C}{2} \]

### Step 4: Substitute Back
Now we substitute back \( z = 2t - 4 \):
\[ -\frac{1}{6} (2t - 4)^{-3} + C \]

### Final Answer
Therefore, the final answer for the integral is:
\[ \int (2t - 4)^{-4} dt = -\frac{1}{6} (2t - 4)^{-3} + C \]