Integrate the function: $f^{\prime}(ax+b)[f(ax+b)]^n$

To integrate $I = \int f^{\prime}(ax+b)[f(ax+b)]^n \, dx$,we use the method of substitution.
Let $u = f(ax+b)$.
Then,differentiating with respect to $x$,we get $du = f^{\prime}(ax+b) \cdot a \, dx$,which implies $f^{\prime}(ax+b) \, dx = \frac{1}{a} \, du$.
Substituting these into the integral:
$I = \int u^n \cdot \frac{1}{a} \, du$
$I = \frac{1}{a} \int u^n \, du$
Using the power rule for integration $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ (where $n \neq -1$):
$I = \frac{1}{a} \cdot \frac{u^{n+1}}{n+1} + C$
Substituting $u = f(ax+b)$ back into the expression:
$I = \frac{[f(ax+b)]^{n+1}}{a(n+1)} + C$