Trigonometric Identities 1 cos x 1 csc x = sin x sin x tan x = cos x cos x cot x = sin x sin2 x + cos2 x = 1 sec x = sec2 x = 1 + tan2 x csc2 x = 1 + cot2 x sin(x ± y) = sin x cos y ± cos x sin y cos(x ± y) = cos x cos y ∓ sin x sin y 1 1 sin x sin y = cos(x − y) − cos(x + y) 2 2 1 1 cos x cos y = cos(x − y) + cos(x + y) 2 2 1 1 sin x cos y = sin(x − y) + sin(x + y) 2 2 sin 2x = 2 sin x cos x cos 2x = cos2 x − sin2 x cos 2x = 1 − 2 sin2 x cos 2x = 2 cos2 x − 1 1 − cos 2x sin2 x = 2 1 + cos 2x 2 cos x = 2 Derivatives d [f (x)g(x)] = f (x)g 0 (x) + g(x)f 0 (x) dx d f (x) g(x)f 0 (x) − f (x)g 0 (x) = dx g(x) [g(x)]2 d [f (g(x))] = f 0 (g(x))g 0 (x) dx d n [x ] = nxn−1 dx d x [e ] = ex dx d x [a ] = ax ln a dx 1 d [ln x] = dx x d [sin x] = cos x dx d [cos x] = − sin x dx d [tan x] = sec2 x dx d [csc x] = − csc x cot x dx d [sec x] = sec x tan x dx d [cot x] = − csc2 x dx d 1 [sin−1 x] = √ dx 1 − x2 d 1 [tan−1 x] = 2 dx x +1 −1 d [cos−1 x] = √ dx 1 − x2 −1 d [cot−1 x] = 2 dx x +1 d 1 √ [sec−1 x] = dx |x| x2 − 1 d −1 √ [csc−1 x] = dx |x| x2 − 1 Integrals Z xn+1 + C if n 6= −1 xn dx = n+1 Z 1 dx = ln |x| + C x Z ex dx = ex + C Z ax ax dx = +C ln a Z sin x dx = − cos x + C Z cos x dx = sin x + C Z sec2 x dx = tan x + C Z csc2 x dx = − cot x + C Z sec x tan x dx = sec x + C Z csc x cot x dx = − csc x + C Z tan x dx = − ln | cos x| + C = ln | sec x| + C Z cot x dx = ln | sin x| + C Z sec x dx = ln | sec x + tan x| + C Z Z Z Z Z Z Z csc x dx = ln | csc x − cot x| + C Z u dv = uv − v du 1 x dx = sin−1 + C a a2 − x2 1 1 x dx = tan−1 + C x2 + a2 a a x 1 1 √ dx = sec−1 + C a a x x2 − a 2 Z n−1 1 sinn−2 x dx sinn x dx = − sinn−1 x cos x + n n Z 1 n−1 n n−1 cos x dx = cos x sin x + cosn−2 x dx n n √ Power Series sin x = ∞ X (−1)n 2n+1 x (2n + 1)! n=0 cos x = ∞ X (−1)n 2n x (2n)! n=0 ex = ∞ X 1 n x n! n=0 ln(1 + x) = IOC = (−∞, ∞) IOC = (−∞, ∞) IOC = (−∞, ∞) ∞ X (−1)n−1 n x n n=1 IOC = (−1, 1] ∞ X 1 = xn 1 − x n=0 tan−1 x = IOC = (−1, 1) ∞ X (−1)n 2n+1 x 2n + 1 n=0 IOC = [−1, 1]