To solve the integral \( \int x^5 f(x^3) \, dx \) given that \( \int f(x) \, dx = g(x) \), we will use integration by parts and substitution. Here’s the step-by-step solution:
### Step 1: Set Up the Integral
We start with the integral:
\[
I = \int x^5 f(x^3) \, dx
\]
### Step 2: Substitution
Let’s use the substitution \( t = x^3 \). Then, we differentiate:
\[
dt = 3x^2 \, dx \quad \Rightarrow \quad dx = \frac{dt}{3x^2}
\]
Also, since \( t = x^3 \), we have \( x = t^{1/3} \) and \( x^2 = (t^{1/3})^2 = t^{2/3} \).
### Step 3: Rewrite the Integral
Now, substitute \( x^5 \) and \( dx \) into the integral:
\[
I = \int (t^{1/3})^5 f(t) \cdot \frac{dt}{3t^{2/3}} = \int \frac{t^{5/3} f(t)}{3t^{2/3}} \, dt
\]
This simplifies to:
\[
I = \frac{1}{3} \int t^{5/3 - 2/3} f(t) \, dt = \frac{1}{3} \int t^{1} f(t) \, dt
\]
### Step 4: Integration by Parts
Now we apply integration by parts, where we let:
- \( u = t \) and \( dv = f(t) \, dt \)
- Then, \( du = dt \) and \( v = g(t) \) (since \( \int f(t) \, dt = g(t) \))
Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \):
\[
I = \frac{1}{3} \left( t g(t) - \int g(t) \, dt \right)
\]
### Step 5: Substitute Back
Now substitute back \( t = x^3 \):
\[
I = \frac{1}{3} \left( x^3 g(x^3) - \int g(x^3) \, dx \right)
\]
### Step 6: Final Expression
Thus, the final expression for the integral \( \int x^5 f(x^3) \, dx \) is:
\[
I = \frac{1}{3} \left( x^3 g(x^3) - \int g(x^3) \, dx \right) + C
\]
where \( C \) is the constant of integration.
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