Quick Solution

Given:

$\int f(x)\, dx = g(x)$

Required: $\int x^5 f(x^3)\, dx$

Use substitution:

Let $u = x^3 \Rightarrow du = 3x^2\, dx \Rightarrow dx = \frac{du}{3x^2}$

Now rewrite the integral:

$$\int x^5 f(x^3)\, dx = \int x^5 f(u) \cdot \frac{du}{3x^2} = \frac{1}{3} \int x^3 f(u)\, du$$

But $x^3 = u$, so:

$$\frac{1}{3} \int u f(u)\, du$$

Now integrate by parts or use the identity:

$$\int u f(u)\, du = u g(u) - \int g(u)\, du$$

Final answer:

$$\int x^5 f(x^3)\, dx = \frac{1}{3} \left[ x^3 g(x^3) - \int g(x^3) \cdot 3x^2\, dx \right] = x^3 g(x^3) - \int x^2 g(x^3)\, dx$$

$$\boxed{ \int x^5 f(x^3)\, dx = x^3 g(x^3) - \int x^2 g(x^3)\, dx }$$