OK, I just did question two, and will attempt the first soon.
y = dx/(x^2 + 2x + 8)
= dx(x^2 + 2x + 1 + 7)
= dx((x + 1)^2 + 7)
this looks a lot like the derivative of an inverse tan function. differentiating...
d(INVtan (x + 1)/7^0.5)/dx
u = x + 1
d(INVtan u/7^0.5)/dx
= 7^0.5 / (7 + u^2) * du/dx
= 7^0.5 / (7 + u^2) * 1 (as du/dx = 1)
= 7^0.5 / (7 + (x + 1)^2)
thus, as d(INVtan (x + 1)/7^0.5)/dx = 7^0.5 / (7 + (x + 1)^2)
y = 1/(x^2 + 2x + 8) = 1/7^0.5 * 7^0.5 / (7 + (x + 1)^2)
= 1/7^0.5 * d(INVtan (x + 1)/7^0.5)/dx
thus
ydx (indefinite integral of y) = 1/7^0.5 * d(INVtan (x + 1)/7^0.5)/dx * dx
= 1/7^0.5 (INVtan (x + 1)/7^0.5)) + c
Please draw attention to any mistakes in my work, as I am rather prone to that