1) INT 1/[x[(4+x^2)^2](1+x^2)] dx

On an earlier problem (on one of the other posts) I explained how to use "Partial Fractions" to solve a particular integration. This one involves the same principle:

Set up the Partial Fractions as:

INT [ A/x + (Bx + C)/(4 + x^2) + (Dx + E)/(4 + x^2)^2 + (Fx + G)/(1 + x^2) ] dx

Find the common denominator for each and add the terms to get the following numerator:

A(4 + x^2)^2*(1 + x^2) + (Bx + C)(x)(4 + x^2)(1 + x^2) + (Dx + E)(x)(1 + x^2) + (Fx + G)(x)(4 + x^2)^2

That's a lot to multiply out, but it can be done (and I don't feel like doing it ). Once you get it done, you'll have a bunch of terms of the following forms: x^6, x^5, x^4, x^3, x^2, x, and Constants. Set the sums of those terms equal to the following:

{terms of x^6} = 0

{terms of x^5} = 0

{terms of x^4} = 0

{terms of x^3} = 0

{terms of x^2} = 0

{terms of x} = 0

{constants} = 1

Then solve for each letter, and use those to solve the above integration (using whatever method is necessary). That's just too much work to type up... so I won't do it unless you specifically ask.