Some of these I cannot solve, but I'll try those that I can:

2) INT sinx/(1+sinx) dx

Multiply the numerator and denominator by 1 - sinx

INT [sinx(1 - sinx)]/(1 - sin^2x) dx

INT (sinx - (1 - cos^2x))/cos^2x dx

INT sinx/cos^2x dx - INT 1/cos^2x dx + INT cos^2x/cos^2x dx

INT secx*tanx dx - INT sec^2x dx + INT 1 dx

secx - tanx + x + C

4) INT x^2(1 + lnx) dx

Let u = 1 + lnx <--> du = 1/x dx

Let dv = x^2 dx <--> v = 1/3*x^3

(1 + lnx)(1/3*x^3) - 1/3 INT (x^3)(1/x) dx

1/3x^3(1 + lnx) - 1/3(1/3)x^3 + C

1/3x^3(1 + lnx) - 1/9x^3 + C

I'll try some of the others in a little bit