MAT 1332 Lecture Notes - Lecture 2: Power Rule, General Idea, Partial Fraction Decomposition

• School: University of Ottawa
• Department: Mathematics
• Course: MAT 1332
• Professor: Robert Smith

Document Summary

Mat1332: calculus for the life sciences ii - part 1. Hint: (cid:90) (cid:90) xndx = (cid:90) (cid:90) xn+1 n + 1. + c (cid:90) (cid:90) 1 x exdx = ex + c dx = ln|x| + c sin xdx = cos x + c cos xdx = sin x + c (cid:90) (cid:90) 1 + y2 dy = arctan y + c arctan (tan x) = x. Then, di erentiating both sides with respect to x, we have. Chain rule : d dx d dy d dy d dy d dy d dy d dy d dy (cid:90) d dx x. = 1 arctan y (1 + y2) = 1 arctan y = 1 + y2 dy = arctan y + c. 1. 3 substitution d dx f (g(x)) = f(cid:48)(g(x))g(cid:48)(x) f (g(x)) = f(cid:48)(g(x))g(cid:48)(x)dx substitute u = g(x) = g(cid:48)(x) du g(cid:48)(x) du dx dx = then f (g(x)) = f(cid:48)(u)g(cid:48)(x) du g(cid:48)(x) f(cid:48)(u)du (cid:90) (cid:90) (cid:90)