3) INT(3x^2 + 3)/[(x-1)^3](x^2 + 1)dx???
4) INT(sinx)/[(cos^3(2x))^1/2]dx???
This one is easy:
INT (3x^2 + 3)/[(x-1)^3(x^2 + 1)] dx
We need to use "Partial Fractions"
INT [A/(x - 1) + B/(x - 1)^2 + C/(x - 1)^3 + (Dx + E)/(x^2 + 1)] dx
Now we need to find the values of A, B, C, D, and E.
A(x - 1)^2(x^2 + 1) + B(x - 1)(x^2 + 1) + C(x^2 + 1) + (Dx + E)(x - 1)^3 = 3x^2 + 3
Multiply all of that out, and match up the x^4, x^3, x^2, x and constant terms. Set the sum of coresponding terms to the following:
{x^4 terms} = 0
{x^3 terms} = 0
{x^2 terms} = 3x^2
{x terms} = 0
{constants} = 3
This represents a system of equations, which you can use to solve for A, B, C, D, and E. Once you have these, solve the integration:
INT [A/(x - 1) + B/(x - 1)^2 + C/(x - 1)^3 + (Dx + E)/(x^2 + 1)] dx
Which at this point, should be much easier to do.