Question

\[\int\frac{\sin^3 x - \cos^3 x}{\sin^2 x \cos^2 x} dx\]

Answer 1

\[\int\left( \frac{\sin^3 x - \cos^3 x}{\sin^2 x \cdot \cos^2 x} \right)dx\]
\[ = \int\frac{\sin^3 x}{\sin^2 x \cdot \cos^2 x}dx - \int\frac{\cos^3 x}{\sin^2 x \cdot \cos^2 x}dx\]
\[ = \int\frac{\sin x}{\cos^2 x}dx - \int\frac{\cos x}{\sin^2 x}dx\]
\[ = \int\frac{\sin x}{\cos x} \times \frac{1}{\cos x}dx - \int\frac{\cos x}{\sin x} \times \frac{1}{\sin x}dx\]
`=∫ sec x  tan x  dx - ∫  "cosec"  x  cot x  dx`
\[ = \sec x - \left( - \text{cosec  x} \right) + C\]
\[ = \sec x + \text{cosec x }+ C\]