`intx^(6) dx = ?`

Options

A. `7x^(7) +C`
B. `(x^(7))/(7) + C`
C. `6x^(5) + C`
D. `6x^(7) + C`

Answer

To solve the integral \( \int x^6 \, dx \), we will use the power rule for integration. The power rule states that:

\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]

where \( n \) is any real number except \(-1\), and \( C \) is the constant of integration.

### Step-by-step Solution:

1. **Identify \( n \)**: In our case, \( n = 6 \).

2. **Apply the power rule**: According to the power rule, we will increase the exponent by 1 and divide by the new exponent:
\[ \int x^6 \, dx = \frac{x^{6+1}}{6+1} + C \]

3. **Calculate the new exponent**:
\[ 6 + 1 = 7 \]

4. **Substitute back into the equation**:
\[ \int x^6 \, dx = \frac{x^7}{7} + C \]

5. **Final answer**: Thus, the integral of \( x^6 \) with respect to \( x \) is:
\[ \int x^6 \, dx = \frac{x^7}{7} + C \]

### Summary of the Solution:
\[ \int x^6 \, dx = \frac{x^7}{7} + C \]