To solve this problem, let's work through the integral provided:

$\int \sin^3 x \cos^4 x \, dx = -\frac{1}{5} (\cos x)^m + \frac{1}{7} (\cos x)^n + c$

where we need to find values of $m$ and $n$ and then determine $m + n$.

Step 1: Simplify the Integral

We start with: $\int \sin^3 x \cos^4 x \, dx$

Since we have a power of $\sin x$ and $\cos x$ in the integral, a common approach is to express $\sin x$ or $\cos x$ in terms of each other using a trigonometric identity. Here, we can use the substitution $u = \cos x$, which implies $du = -\sin x \, dx$.

Rewrite $\sin^3 x$ and $\cos^4 x$ in Terms of $u$

1. Since $\sin^2 x = 1 - \cos^2 x = 1 - u^2$, we get: $\sin^3 x = \sin^2 x \cdot \sin x = (1 - u^2)(-\frac{du}{\sin x})$