Integral Calculus: integrals (of, say, csc(3x+8)cot(3x+8)) w/o U-Substitution

[bachsci]

I've been given a problem set that seems to obviously need substitution (which I know how to use) but as my prof has yet to cover it, we aren't allowed to use it for the given problems. The biggest one is the integral of csc(3x+8)cot(3x+8). I know that csc(x)cot(x) is a known integral, being -csc(x) but it doesn't seem possible to do this question without substituting for 3x+8.

I've tried changing the trig functions to 1/sin^2(3x+8) and cos(3x+8) and using integration by parts but there still comes a point where I need substitution. The answer is
-(1/3)csc(3x+8)+C but I don't know what method to use to support it.

 

[Deleted member 4993]

I've been given a problem set that seems to obviously need substitution (which I know how to use) but as my prof has yet to cover it, we aren't allowed to use it for the given problems. The biggest one is the integral of csc(3x+8)cot(3x+8). I know that csc(x)cot(x) is a known integral, being -csc(x) but it doesn't seem possible to do this question without substituting for 3x+8.

I've tried changing the trig functions to 1/sin^2(3x+8) and cos(3x+8) and using integration by parts but there still comes a point where I need substitution. The answer is
-(1/3)csc(3x+8)+C but I don't know what method to use to support it.

There is one silly way (as far as I am concerend:

d(3x+8) = d(3x) = 3 dx → dx = 1/3 * d(3x+8)

Then

\(\displaystyle \displaystyle{\int csc(3x+8)*cot(3x+8) dx}\)


\(\displaystyle \displaystyle{= \ \int csc(3x+8)*cot(3x+8)* \dfrac{1}{3} d(3x+8)}\)

and continue....