Using a suitable substitution, or otherwise, find the integral of [x/((7+2*(x^2))^2)].

The point of using a substitution when dealing with integrals is to reduce the problem from a very complicated integral to a simplistic one. How can we achieve this with our integral? Well, we know that when we use substitutions we will be replacing our dx term with a du term. i.e., if we choose u=x for our substitution, then we replace dx with du in our original integral and proceed as normal. This is the same as if we chose u=3+x, as when differentiating with respect to x then the constant term would disappear. Well, for our integral, the complicated part is the equation in the brackets on the denominator, (7+2*(x^2))^2. The numerator is just an x, so we also need a substitution that will replace our dx with a du/x, so that these x terms would cancel out.Well, if we try to make the entire denominator our substitution, i.e. u=(7+2*(x^2))^2, then du/dx = 8x*(7+2*(x^2)), so dx=du/(56x+16x^2). This isn't right for us, as we need dx=du/x to cancel with our numerator x. So what if we try again, but this time make our substitution the square root of the denominator, i.e. just the terms in the bracket, so that u=7+2*(x^2). Then we have du/dx = 4x. Our integral then becomes the integral of x/(u^2) * du/(4x). This works, since we can cancel our numerator x with our denominator x, making the integral become the integral of 1/(4*(u^2)) du. This is just 1/4 of the integral of u^(-2), which is 1/4 of -u^(-1). Then our answer is simply -(1/(4u)). Reversing our substitution, the answer of the original integral in terms of x is then -(1/(4*(7+2*(x^2))). And we are finished!