Question

Find the integrals of the function: $\frac {sin^3 x+cos^3 x}{sin^2 x .cos^2 x}$

Answer 1

$\frac {sin^3 x+cos^3 x}{sin^2 x .cos^2 x}$

= $\frac {sin^3 x}{sin^2 x .cos^2 x}$ + $\frac {cos^3 x}{sin^2 x .cos^2 x}$

= $\frac {sin \ x}{cos^2 x }$+ $\frac {cos\ x}{sin^2 x }$

= $tan \ x .sec\ x + cot \ x. cosec\ x$

∴ $\int\frac {sin^3 x+cos^3 x}{sin^2 x .cos^2 x}dx$ = $\int (tan \ x .sec\ x + cot \ x. cosec\ x)dx$
= $sec\ x-cosec\ x +C$