Step-by-step Solution:

We start with the given expression: \[ x^3 = t \] Differentiating both sides: \[ 3x^2 dx = dt \quad \text{or} \quad x^2 dx = \frac{dt}{3} \] Now, we focus on the integral \( I = \int x^5 f(x^3) \, dx \). \[\] We can rewrite the integral as follows: \[ I = \int x^2 \cdot x^3 f(x^3) \, dx \] Next, we substitute \( x^3 = t \), and \( 3x^2 dx = dt \). So we have: \[ I = \frac{1}{3} \int t f(t) \, dt \] Now, we apply integration by parts. Recall that the formula for integration by parts is: \[ \int u \, dv = uv - \int v \, du \] Here, let: \[\] \( u = t \) \[\] \( dv = f(t) \, dt \) \[\] Thus: \[\] \( du = dt \) \[\] \( v = \int f(t) \, dt \) Using the integration by parts formula: \[ I = \frac{1}{3} \left[ t \int f(t) \, dt - \int \left( \frac{dt}{dt} \int f(t) \, dt \right) dt \right] \] Simplifying this: \[ I = \frac{1}{3} \left[ t \cdot g(t) - \int g(t) \, dt \right] \] where \( g(t) = \int f(t) \, dt \). \[\] Now, substituting \( t = x^3 \) and \( dt = 3x^2 dx \), we get: \[ I = \frac{1}{3} x^3 g(x^3) - \frac{1}{3} \int x^2 g(x^3) \, dx + c \] Thus, the final result is: \[ I = \frac{1}{3} x^3 g(x^3) - \int x^2 g(x^3) \, dx + c \]