Methods in calculus

• An integral is improper if one or both limits are infinite or the function is undefined at some point.
• If an indefinite integral exists, it is said to converge. If it doesn't, it is said to diverge.
• To evaluate an integral with one infinite limit, replace the infinity with another variable (t is usually used), evaluate as usual, then consider the limit of the result as t -> infinity.
• To evaluate an integral with both limits infinite, split up the integral into two parts and evaluate each part separately then add them. Have one part between k and infinity and the other between negative infinity and k, where k is any integer (usually 0).
• To evaluate an integral that is undefined at a point, identify which point is undefined and replace with any variable (usually t), splitting the integral up into two parts if necessary, having one between a and t and the other between t and b, where a and b are the limits of the integral. Evaluate, then consider the result as t -> the undefined point.
• Wherever it is required to split up an integral, both parts have to converge in order for the overall integral to converge. If any one diverges, the whole interval diverges.
• The mean value of a function over the interval a to b is 1/(b - a) multiplied by the integral between a and b of the function.
• The mean value, F of a function f(x) + k is F + k, k f(x) is k F and -f(x) is -F.
• The differential of arcsin x is 1/ root(1 - x^2).
• The differential of arccos x is -1 / root(1 - x^2).
• The differential of arctan x is 1/ 1 + x^2.
• The integral of 1/(a^2 + x^2) is 1/a arctan (x/a) + c.
• The integral of 1/root(a^2 - x^2) dx is arcsin (x/a) + c.
• The integral of 1/(a^2 - x^2) dx is 1/2a ln ((a + x)/(a - x)) + c and to prove it the integral can be written as 1/(a + x)(a - x) and then split into partial fractions.
• For integrals involving a^2 + x^2 try the substitution x = a tan u.
• For integrals involving root(a^2 - x^2), try the substitution x = a sin u.
• The volume of revolution formed when a function y = f(x) is rotated through 2 pi about the x axis over the range x = [a, b] is V = pi multiplied by the integral between a and b of y^2.
• The volume of revolution formed when a function y = f(x) is rotated through 2 pi about the y axis over the range y = [a, b] is V = pi multiplied by the integral between a and b of x^2.
• The volume of a cylinder is V = pi r^2 h.
• The volume of a cone is V = (pi r^2 h)/3
• The integral volume of revolution of a parametrically defined curve around the x axis is y^2 dx/dt dt.
• The integral for a volume of revolution of a parametrically defined curve around the y axis is x^2 dy/dt dt.