Integral of (x<sup>2</sup> + 1)<sup>1/2</sup> is a substitution problem, let x = tan@
Integral of cot x / ln(esin x) - an interesting problem. Use log laws to rewrite the denominator as 1 - ln(cosec x), then use the substitution u = cosec x to transform the problem to int 1 / [u(ln u - 1) du. A second substitution, v = ln u, gives int 1 / (v - 1) dv. Thus, the answer is ln |v - 1| + C = ln |ln u - 1| + C = ln |ln(sin x) + 1| + C
Integral of 1 / (2x - x<sup>2</sup>)<sup>1/2</sup>, complete the square on x, and rewrite as 1 / sqrt[1 - (x - 1)<sup>2</sup>]. It integrates to an inverse trig function.