1) Compute the indefinite integral: integral 2x ln(x^2 - 8x + 18) dx
2) Note that x^4 + 1 = (x^4 + 2x^2 + 1) = (x^2 + 1)^2 - (sqrt2x)^2 ... use this trick to compute the following indefinite integral as an elementary function (with exact coefficients).
integral 1 / (x^4 + 2x^2 + 4) dx
3) Prove analytically that there is an elementary function F(x) such that
F'(x) = 1 / (x^7 - 1)
HINT: use the result of the Fundamental Theorem of Algebra, and carry on abstractly.
4) Prove the following: Let (ax^2 + bx + x)^n be an n-th power of an irreducible quadratic polynomial with a positive integer n, and let S(x) be a polynomila with degree (S(x)) < 2n. Then, there are constants A1,....,An, and B1,...,Bn, such that:
S(x)/(ax^2+bx+c)^n = (A1x +B1)/(ax^2+bx+c)^1 + (A2x + B2)/(ax^2+bx+c)^2 +...+
(Anx+Bn)/(ax^2+bx+c)^n