Concept:\(\displaystyle \int \ x^n\ dx = \frac{x^{n+1}}{n+1}+C, n\neq -1\)Calculation:\(\displaystyle I=\int (\sin x)^{-1/2} (\cos x)^{-3/2}dx \)? \(\displaystyle I=\int \frac{1}{(\sqrt{sin \ x.cos^3\ x})}dx \)? \(\displaystyle I=\int \frac{1}{cos\ x\sqrt{sin \ x.cos\ x}}dx \)Multiplying and dividing cosec2x in numerator and denominator, we get,? \(\displaystyle I=\int \frac{cosec^2x}{cosec^2x.cos\ x\sqrt{sin \ x.cos\ x}}dx \)? \(\displaystyle I=\int \frac{cosec^2x}{\frac{cos\ x}{sin\ x}\sqrt{\frac{sin \ x.cos\ x}{sin^2x}}}dx \)? \(\displaystyle I=\int \frac{cosec^2x}{cot\ x\sqrt{cot\ x}}dx \)Let t = cot x ? dt = - cosec2x? \(\displaystyle I=\int \frac{-dt}{t^\frac{3}{2}}\)? \(\displaystyle I=\int -t^{\frac{-3}{2}}dt=\frac{-t^{\frac{-1}{2}}}{\frac{-1}{2}}\)? \(\displaystyle I= \frac{2}{\sqrt{t}}+C\)? \(\displaystyle I= \frac{2}{\sqrt{cot\ x}}+C\)? \(I=2\sqrt{tan\ x}+C\)? \(\displaystyle \int (\sin x)^{-1/2} (\cos x)^{-3/2}dx =2\sqrt{tan\ x}+C\)