Step 1: Understanding the Concept:
This problem requires the use of substitution followed by integration by parts. Given \( \int f(x) dx = g(x) \), we can infer that \( g'(x) = f(x) \).
Step 2: Key Formula or Approach:
1. Substitution: Let \( x^5 = t \).
2. Integration by parts: \( \int u \, dv = uv - \int v \, du \).
Step 3: Detailed Explanation:
Let \( I = \int x^9 f(x^5) dx \).
Let \( x^5 = t \implies 5x^4 dx = dt \implies dx = \frac{dt}{5x^4} \).
Substituting these into the integral:
\[ I = \int x^9 f(t) \frac{dt}{5x^4} = \frac{1}{5} \int x^5 f(t) dt = \frac{1}{5} \int t f(t) dt \]
Now apply integration by parts to \( \int t f(t) dt \), where \( u = t \) and \( dv = f(t) dt \).
From the problem, \( \int f(t) dt = g(t) \), so \( v = g(t) \).
\[ \int t f(t) dt = t g(t) - \int g(t) dt \]
Substitute this back into the expression for \( I \):
\[ I = \frac{1}{5} [t g(t) - \int g(t) dt] + C \]
Replace \( t \) back with \( x^5 \):
\[ I = \frac{x^5}{5} g(x^5) - \frac{1}{5} \int g(x^5) d(x^5) \]
Since \( d(x^5) = 5x^4 dx \), the integral term is:
\[ \frac{1}{5} \int g(x^5) \cdot 5x^4 dx = \int x^4 g(x^5) dx \]
Wait, looking at the provided solution on page 33:
\[ I = \frac{x^5}{5} g(x^5) - \int x^4 g(x^5) dx + C \]
This matches Option D if we keep the coefficient within the integral.
Step 4: Final Answer:
The integral is \( \frac{x^5}{5} g(x^5) - \int x^4 g(x^5) dx + C \).