To solve the integral \( \int x^5 \, dx \), we will apply 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 the exponent**: In our case, the exponent \( n \) is 5 since we are integrating \( x^5 \).
2. **Apply the power rule**: According to the power rule, we will increase the exponent by 1 and divide by the new exponent. Thus, we have:
\[
n + 1 = 5 + 1 = 6
\]
3. **Write the integral using the power rule**:
\[
\int x^5 \, dx = \frac{x^{6}}{6} + C
\]
4. **Final expression**: Therefore, the result of the integral is:
\[
\int x^5 \, dx = \frac{x^6}{6} + C
\]
### Conclusion:
The integral \( \int x^5 \, dx \) is equal to \( \frac{x^6}{6} + C \).
To solve the integral \( \int x^5 \, dx \), we will apply 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.
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