hi, is there any method of integrating a hyperbolic or trignometric function whos degree is 4, such as (tanx)^4, (cosechx)^4 WITHOUT resorting to reduction formulae??
Sure, there just isn't a "golden recipe" which will be succesfull every time. You'll need to play arround with trig formulas. I'll do one as an example.
Now I'll split the integral in two. The first one is easy since the derivative of tan(x) is exactly secē(x), so:
For the second one; I convert to sin(x) and cos(x):
Splitting the integral in two again gives us simply:
So we conclude, without reduction-formula: