Integration , how do I pick u ?Watch
Also these 2:
u = x^2
seems a sensible choice as when you differentiate it a couple of times, it becomes a constant and the integral is much easier. That means the other term which you integrate is
v = sec^2(x)tan(x)
for the first one. That is a standard derivative of tan^2(x), so integrating is relatively easy and integrating tan^2(x) another time isn't too hard.
So that choice of u and v is sensible. Similar for the other two integrals.
The 12x^2(3+2x)^5 one doesn't really fall under those rules. In this case, the power of the bit you integrate will go up by 1, and the power of the bit you differentiate will go down by 1. Since you're trying to reduce one of them to a constant, you want to differentiate the part where the power is smaller.
However, to be honest in this case if I just had to find the integral I'd probably expand the whole thing out in powers of x and integrate. Or rewrite 12x^2 as 3 ((2x+3)-3)^2 = 3((2x+3)^2 - 6(2x+3) + 9) to expand as powers of (2x+3) and then integrate. Don't neglect the possibilitity of just "doing some algebra" to get to something you can integrate.