`int (2x+7)^6` dx is equal to:

Options

A. `(2x+7)^7/14+c`
B. `(2x+7)^6/14+c`
C. `(2x+7)^7/7+c`
D. `(-(2x+7)^7)/14+c`

Answer

To solve the integral \( \int (2x + 7)^6 \, dx \), we will use the substitution method. Here are the steps:

### Step 1: Substitution
Let \( t = 2x + 7 \).

### Step 2: Differentiate
Now, differentiate \( t \) with respect to \( x \):
\[ \frac{dt}{dx} = 2 \implies dt = 2 \, dx \implies dx = \frac{dt}{2} \]

### Step 3: Rewrite the Integral
Substituting \( t \) and \( dx \) into the integral:
\[ \int (2x + 7)^6 \, dx = \int t^6 \cdot \frac{dt}{2} \]
This simplifies to:
\[ \frac{1}{2} \int t^6 \, dt \]

### Step 4: Integrate
Now, we can integrate \( t^6 \):
\[ \int t^6 \, dt = \frac{t^{7}}{7} + C \]
Thus,
\[ \frac{1}{2} \int t^6 \, dt = \frac{1}{2} \cdot \left( \frac{t^{7}}{7} + C \right) = \frac{t^{7}}{14} + \frac{C}{2} \]

### Step 5: Substitute Back
Now, substitute back \( t = 2x + 7 \):
\[ \frac{(2x + 7)^{7}}{14} + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int (2x + 7)^6 \, dx = \frac{(2x + 7)^{7}}{14} + C \]