The given function is \( \displaystyle \frac{5\cos^{3}x + 7\sin^{2}x}{\sin^{2}x \cdot \cos^{2}x} \) and we need to find the integration of this function with respect to \( x \).
Solution:
\( \begin{align*} &\int \frac{5\cos^{3}x + 7\sin^{2}x}{\sin^{2}x \cdot \cos^{2}x} \, dx \\ &= \int \left( \frac{5\cos^{3}x}{\sin^{2}x \cdot \cos^{2}x} + \frac{7\sin^{2}x}{\sin^{2}x \cdot \cos^{2}x} \right) dx \\ &= \int \left( \frac{5\cos x}{\sin^{2}x} + \frac{7}{\cos^{2}x} \right) dx \\ &= 5 \int \frac{\cos x}{\sin^{2}x} dx + 7 \int \sec^{2}x \, dx \\ &= 5 \int \frac{\cos x}{\sin x} \cdot \frac{1}{\sin x} dx + 7 \int \sec^{2}x \, dx \\ &= 5 \int \cot x \cdot \operatorname{cosec}x \,dx + 7 \int \sec^{2}x \, dx \\ &= 5 (- \operatorname{cosec}x) + 7 \tan x + C, \quad \text{C is a constant} \\ &= -5 \operatorname{cosec}x + 7 \tan x + C \\ \end{align*} \)Thus, the integration of the given function is
\( \boxed{ \int \left( \frac{5\cos^{3}x}{\sin^{2}x \cdot \cos^{2}x} + \frac{7\sin^{2}x}{\sin^{2}x \cdot \cos^{2}x} \right) dx = -5 \operatorname{cosec}x + 7 \tan x + C } \\ \)Please let me know in the comments if you find any errors in this solution.