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Pattern - 22 of Indefinite Integration

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⋆ Pattern – 22

1) $I=\ \int\frac{1}{\sin^\frac{3}{2}{x}\times \cos^\frac{1}{2}{x}}\ dx$

= $\int\frac{1}{\left(\frac{\sin{x}}{\cos{x}}\right)^\frac{3}{2}\times\ \left(\cos{x}\right)^{\frac{3}{2}+\frac{1}{2}}}\ dx$

= $\ \int\frac{1}{\tan^\frac{3}{2}{x}\times \cos^2{x}}dx$

= $\ \int\frac{\sec^2{x}}{\tan^\frac{3}{2}{x}}\ dx$

Let tan x = y ∴ $\sec^2{x} dx$ = dy

= $\ \int\frac{1}{y^\frac{3}{2}}\ dy$

= $\ \int y^\frac{-3}{2}dy$

= $\ \frac{y^\frac{7}{2}}{\frac{-1}{2}}+c$

= $\ \frac{-2}{y^\frac{1}{2}}+c$

= $\ \frac{-2}{\tan^\frac{1}{2}{x}}+c$

= $\ \frac{-2}{\sqrt{\tan{x}}}+c$