[TEX]\int \frac{sin^2x}{cos^6x}dx =\int \tan^2x.\frac{dx}{cos^4x}=\int tan^2x.(1+tan^2x)d(tanx)
\int \frac{dx}{sinx}=\int \frac{dx}{2sin\frac{x}{2}cos\frac{x}{2}}=\int \frac{dx}{2tan\frac{x}{2}cos^2\frac{x}{2}}=\int \frac{d(tan\frac{x}{2})}{tan\frac{x}{2}}=ln\left | tan\frac{x}{2} \right [/TEX]