kindly check... this is an integral without using the formula of trig integrals:

$\displaystyle \int \cot(5x) + \csc(5x) dx$

solution:

$\displaystyle \int \frac{\cos(5x)}{\sin(5x)} + \csc(5x) dx$

from the first integral

let u = sin(5x)

du = cos(5x)dx

from the right integral u multiply both denominator and numerator by (cot(5x) + csc(5x))

$\displaystyle \frac{1}{5}\ln{(\sin{5x})} + \int \frac{\csc(5x)\cot(5x) + \csc^2(5x)}{\cot(5x)+\csc(5x)} dx$

let u = $\displaystyle \cot(5x)+\csc(5x)$

du = $\displaystyle -5(\csc^2(5x) + \csc(5x)\cot(5x)) dx$

integrated to :

$\displaystyle \frac{1}{5}\ln{(\sin{5x})} - \frac{1}{5}\ln(\cot(5x) + \csc(5x))$

i ended up having

$\displaystyle \frac{1}{5} \ln(\frac{\sin^2(5x)}{\cos(5x) + 1}) + C$

the answer at back of book is supposed to be:

$\displaystyle \frac{1}{5} \ln (1-\cos(5x)) + C$

and you shud not use a formula of trig integral