y = sec x + tan x

sec²x – tan²x = 1

 $\mathrm{secx}-\mathrm{tanx}=\frac{1}{\mathrm{y}}$

 $\therefore 2\mathrm{secx}=\mathrm{y}+\frac{1}{\mathrm{y}}\Rightarrow \mathrm{secx}=\frac{{\mathrm{y}}^2+1}{2\mathrm{y}}$

dy = (sec x tan x + sec²x) dx  $\Rightarrow$ dy = y sec x dx $\Rightarrow$ dx =$\frac{\mathrm{dy}}{\mathrm{ysecx}}$

 $\therefore \mathrm{I}=\int \frac{{\sec }^2\mathrm{x}\left(\frac{\mathrm{dy}}{\mathrm{ysecx}}\right)}{{\mathrm{y}}^{9/2}}=\int \frac{\frac{{\mathrm{y}}^2+1}{2\mathrm{y}}}{{\mathrm{y}}^{11/2}}\mathrm{dy}=\frac{1}{2}\int \frac{{\mathrm{y}}^2+1}{{\mathrm{y}}^{13/2}}\mathrm{dy}=\frac{1}{2}\int ({\mathrm{y}}^{-9/2}+{\mathrm{y}}^{-13/2})\mathrm{dy}$

 $\mathrm{I}=\frac{1}{2}(\frac{{\mathrm{y}}^{-7/2}}{-7/2}+\frac{{\mathrm{y}}^{-11/2}}{-11/2})+\mathrm{K}$

 $\mathrm{I}=\frac{-1}{7{(\mathrm{secx}+\mathrm{tanx})}^{7/2}}-\frac{1}{11{(\mathrm{secx}+\mathrm{tanx})}^{11/2}}+\mathrm{K}$

 $\mathrm{I}=\frac{-1}{{(\mathrm{secx}+\mathrm{tanx})}^{11/2}}(\frac{1}{11}+\frac{{(\mathrm{secx}+\mathrm{tanx})}^2}{7})+\mathrm{K}$