Find $h\left(x\right)$ by evaluating the following indefinite integral.
$h\left(x\right)=\int x^{8}dx$

• A. $h\left(x\right)=x^9+C$
• B. $h\left(x\right)=\frac{x^9}{9}+C$
• C. $h\left(x\right)=\frac{x^9}{9}$
• D. $h\left(x\right)=x^9$

Answer: (B)

Verified step by step guidance

1. Identify the integral to be evaluated: \( \int x^8 \, dx \). This is an indefinite integral, meaning we are looking for a function whose derivative is \( x^8 \).
2. Recall the power rule for integration, which states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.
3. Apply the power rule to the integral \( \int x^8 \, dx \). Here, \( n = 8 \), so we increase the exponent by 1 to get \( x^{8+1} = x^9 \).
4. Divide by the new exponent to complete the integration: \( \frac{x^9}{9} \).
5. Don't forget to add the constant of integration \( C \) to the result, giving the final expression for \( h(x) \) as \( h(x) = \frac{x^9}{9} + C \).