Why can't I integrate trigonometric functions without making a substitution? For example, $\int\sin^3x$ is turned into $\int sinxsin^2x$ then a substitution for $\sin^2x$ is made. What I would have done was $\int(sinx)^3$ and integrate via recognition:  $\int (ax+b)^n dx$ =$\frac{(ax+b)^{n+1}}{a(n+1)} + C$ . However this will give a different answer from the correct one. Why is that? Why can't this reverse chain rule work for trigonometric functions?

This is basically boils down to the fact that
$$
(x^n)' = n \, x^{n-1}
$$
but
$$
(\sin^n x)' = n \, \sin^{n-1} x \, \cos x.
$$
You can see how these are not 'analogous' so you can't generalize the integral formula for $(a \, x + b)^n$
