Q.Integrate: $\int \sin^5 x\cos^8 x\,dx$

The integrals of the four "reciprocal-family" trigonometric functions all follow from Corollary III of the substitution theorem ($\int\frac{f'(x)}{f(x)}dx=\log|f(x)|+c$): $\int\tan x\,dx=\log|\sec x|+c$ (write $\tan x=\frac{\sin x}{\cos x}$, numerator is $-1$ times the derivative of the denominator); $\int\cot x\,dx=\log|\sin x|+c$; $\int\sec x\,dx=\log|\sec x+\tan x|+c$ (multiply top and bottom by $\sec x+\tan x$, whose derivative is exactly $\sec x\tan x+\sec^2x$, the new numerator); and $\int\csc x\,dx=\log|\csc x-\cot x|+c$ by the parallel trick. These four results, together with the substitution corollar …