`int(2x+3)^(5)dx=?`

Options

A. `((2x+3)^(6))/(6)+C`
B. `((2x+3)^(4))/(8)+C`
C. `((2x+3)^(6))/(12)+C`
D. none of these

Explanation

To solve the integral \( \int (2x + 3)^5 \, dx \), we can use the method of substitution. Here are the steps:

### Step 1: Choose a substitution
Let \( t = 2x + 3 \).

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

### Step 3: Substitute in the integral
Substituting \( t \) and \( dx \) into the integral gives:
\[ \int (2x + 3)^5 \, dx = \int t^5 \left(\frac{dt}{2}\right) \]
This simplifies to:
\[ \frac{1}{2} \int t^5 \, dt \]

### Step 4: Integrate
Now, we can integrate \( t^5 \):
\[ \int t^5 \, dt = \frac{t^6}{6} + C \]
Thus, we have:
\[ \frac{1}{2} \left( \frac{t^6}{6} + C \right) = \frac{t^6}{12} + \frac{C}{2} \]

### Step 5: Substitute back for \( t \)
Now, substitute back \( t = 2x + 3 \):
\[ \frac{(2x + 3)^6}{12} + C \]

### Final Answer
Thus, the integral \( \int (2x + 3)^5 \, dx \) is:
\[ \frac{(2x + 3)^6}{12} + C \]