Question

$\int \frac{5{\cos }^{3}x+6{\sin }^{3}x}{2{\sin }^{2}x{\cos }^{2}x}dx$

Answer 1

$$\begin{gathered} \int \left(\frac{5{\cos }^{3}x+6{\sin }^{3}x}{2{\sin }^{2}x{\cos }^{2}x}\right)dx \\ =\int \left(\frac{5{\cos }^{3}x}{2{\sin }^{2}x{\cos }^{2}x}+\frac{6{\sin }^{3}x}{2{\sin }^{2}x{\cos }^{2}x}\right)dx \\ =\int \left(\frac{5}{2}\frac{\cos x}{{\sin }^{2}x}+3\frac{\sin x}{{\cos }^{2}x}\right)dx \\ =\frac{5}{2}\int \left(\frac{\cos x}{\sin x}\times \frac{1}{\sin x}\right)dx+3\int \frac{\sin x}{\cos x}\times \frac{1}{\cos x}dx \\ =\frac{5}{2}\int \left(\mathrm{cosec}x\cot x\right)dx+3\int \sec x\tan xdx \\ =\frac{5}{2}\left(-\mathrm{cosec}x\right)+3\sec x+\mathrm{C} \\ =-\frac{5}{2}\mathrm{cosec}x+3\sec x+\mathrm{C} \end{gathered}$$