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Pattern - 23 of Indefinite Integration

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⋆ Pattern – 23

$\int\frac{1}{\sin^2{x}\times \cos^2{x}}\ dx,\ \int\frac{1}{\sin{x}\times \cos^2{x}\ }dx,$

$\int\frac{1}{\sin^2{x}\times \cos{x}}\ dx,\ \int\frac{1}{\sin{x}\times \cos{x}}\ dx,$

∫ $\sec^2{x}\times cosec^2\ x$ dx, ∫ sec x ∙ cosec x dx etc.

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

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

$=\ \int\left(\sec^2{x}+cosec^2x\right)dx$

= tan x – cot x + c

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

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

= ∫ ( sec x ∙ tan x + cosec x ) dx

= sec x + log | cosec x – cot x | + c