1/u = x
Limits become,
0 and Infinity
I u^2/(u^4 + 1) du Limits( Infinity & 0)
I = 1/2Arctan(u^2) * u - I
2I = Infinity
I = Infinity
EDIT: Are we allowed to use the hint? Partial Fractions become very difficult but eventually yield the same answer.
Ok, For the first question.
E: x^2/16 + y^2/9 = 1
Tangents are drawn from P(3, 4)
If you substitute x=3, you will get two answers. Same for y, Find the gradient and then the first questions done. (Two points is probably the only way) Didn't come out either.
The second,
Didn't come out...
The conics question is simple, just sub in either the X or Y value and solve the Quadratic.
The differentiation, Do you know implicit differentiation?
The graphs, that just means that the graph is half as wide as before.
There is no way that you can 'understand' string theory without knowing the maths behind it. There is a pretty general description on Wikipedia, however if you are really interested buy a intro to String Theory book from Amazon.