I am having trouble with a few areas in my calc 2 study. I do not completely understand the steps involved in my book, and I would like to fix this before my final tomorrow. (btw is latex working yet?)

For example in the integral, int[ (4x^2+9)^(1/2) / x^4 ] dx

I do not understand how to approach the problem at all. I know it is a trigonometric substitution, but when I look at my teachers notes I see she has the equation int[ (3*sec(t) ( 3/2 sec^2 (t)) dt) / ((3^4/2^4) (tan^4 (t))) ]dt as the next step.

I have no idea where the (3/2 sec^2 (t)) came from. It looks like the first term (3*sec(t)) only integrated, but i know it hasn't been, because that isn't the integrated form of 3*sec(t). I also have no idea how she got her denominator.

3^x/2^x * tan^x (t) ???

Any help here or on trig subs in general would be nice

Similarly on the trig integral: int[ (cos^4(x)) ], i follow it fine until my book states the coefficient of the following step:

=1/4 int[dx] + 1/4 int[ (cos 2*x)(2)(dx) ] + 1/8 int [ (1+cos 4*x) dx ]

where does the 1/8 come from here?

the same problem is seen in the integral int [ sin^2(x) dx ]

on the step = 1/2 int[dx] - 1/4 int [ cos 2x * 2 * dx ], i dont see where the 1/4 is coming from.

thanks.