Tables Of Integrals

Table of Integrals (Antiderivatives) Throughout these tables, a, b and C are constants, and n is a positive integer. f (x) af (x) + bg(x) f (x) + g(x) f (x) − g(x) af (x) 1 xa 1 x x

e eax ax sin x cos x sec2 x csc2 x sec x tan x csc x cot x tan x cot x sec x csc x √ 1 1−x2 1 1+x2

R aR R

R F (x) = f (x) dx R f (x) dx + Rb g(x) dx + C f (x) dx + R g(x) dx + C f (x) Rdx − g(x) dx + C a f (x) dx + C x+C xa+1 a+1 + C if a 6= −1 ln |x| + C ex + C 1 ax +C ae 1 x ln a a + C − cos x + C sin x + C tan x + C − cot x + C sec x + C − csc x + C ln | sec x| + C ln | sin x| + C ln | sec x + tan x| + C ln | csc x − cot x| + C arcsin x + C arctan x + C

Integration by Parts Table f (x) u(x)v 0 (x) xeax xn eax ln(ax) n x ln(ax)

R F (x) = f (x) dx R u(x)v(x) − u0 (x)v(x) dx + C 1 1 ax ax − +C 2e a xe R an−1 1 n ax n ax x e − x e dx + C a a x ln(ax) − x + C 1 1 n+1 n+1 ln(ax) − (n+1) +C 2x n+1 x

x sin(ax) xn sin(ax) x cos(ax) xn cos(ax)

− a1 x cos(ax)R+ a12 sin(ax) + C − a1 xn cos ax + na xn−1 cos(ax) dx + C 1 + a12 cos(ax) + C a x sin(ax) R 1 n n xn−1 sin(ax) dx + C a x sin ax − a
ebx b sin(ax) − a cos(ax) +C 2 2 a +b
ebx a2 +b2 a sin(ax) + b cos(ax) + C √ x arcsin(ax) + a1 1 − a2 x2 + C 1 x arctan(ax) − 2a ln(1 + a2 x2 ) + C

sin(ax)ebx cos(ax)ebx arcsin(ax) arctan(ax)

Derivation method u = x, dv = eax dx u = xn , dv = eax dx u = ln(ax), dv = dx u = ln(ax), dv = xn dx u = x, dv = sin(ax) dx u = xn , dv = sin(ax) dx u = x, dv = cos(ax) dx u = xn , dv = cos(ax) dx twice: u = sin(ax), dv = ebx dx twice: u = cos(ax), dv = ebx dx u = arcsin(ax), dv = dx u = arctan(ax), dv = dx