微积分基本公式.docx

D sin x=cos x x cos x = -sin x tan x = sec 2 x cot x = -csc 2 x sec x = sec x tan x csc x = -csc x cot x x 1 D sin-1 ( )= 微积分公式 sin x dx = -cos x + C cos x dx = sin x + C tan x dx = ln |sec x | + C cot x dx = ln |sin x | + C sec x dx = ln |sec x + tan x | + C csc x dx = ln |csc x – cot x | + C sin-1 x dx = x sin-1 x+ 1 ? x2 +C  sin-1(-x) = -sin-1 x cos-1(-x) = - cos-1 x tan-1(-x) = -tan-1 x cot-1(-x) = - cot-1 x sec-1(-x) = - sec-1 x csc-1(-x) = - csc-1 x x sinh-1 ( )= ln (x+ a2 ? x2 ) x? R x a a2 ? x2 a x cos-1 ( a )= x tan-1 ( )= ?1 a2 ? x2 a cos-1 x dx = x cos-1 x- 1 ? x2 +C tan-1 x dx = x tan-1 x-?ln (1+x2)+C cot-1 x dx = x cot-1 x+?ln (1+x2)+C x cosh-1 ( a )=ln (x+ x2 ? a2 ) x≧1 x 1 a ? x tanh-1 ( )= ln ( ) |x| 1 a a2 ? x2 a 2a a ? x x ?a cot-1 ( a )= a2 ? x2 sec-1 x dx = x sec-1 x- ln |x+ x2 ? 1 |+C x 1 x ? a coth-1 ( a )= 2a ln ( x ? a ) |x| 1 x a csc-1 x dx = x csc-1 x+ ln x ? 1 1 ? x2 sec-1 ( a )= x x2 ? a2 |x+ x2 ? 1 |+C sech-1( a )=ln( x + ≦1 x2 )0≦x x ?a x 1 1 ? x 2 csc-1 ( a )= x  x2 ? a2 csch-1 ( a )=ln( x + x2 ) |x| 0 D sinh x = cosh x x cosh x = sinh x tanh x = sech 2 x coth x = -csch 2 x sech x = -sech x tanh x sinh x dx = cosh x + C cosh x dx = sinh x + C tanh x dx = ln | cosh x |+ C coth x dx = ln | sinh x | + C sech x dx = -2tan-1 (e-x) + C duv = udv + vdu duv = uv = udv + - udv = uv - vdu cos2θ-sin2θ=cos2θ cos2θ+ sin2θ=1  vdu csch x = -csch x coth x csch x dx = 2 ln | 1 ? e? x 1 ? e?2 x | + C cosh2θ-sinh2θ=1 cosh2θ+sinh2θ=cosh2θ xD sinh-1( )= x x a 1 a 2 ? x 2 sinh-1 x dx = x sinh-1 x- 1 ? x2 + C cosh-1 x dx = x cosh-1 x- x2 ? 1 + C sin 3θ=3sinθ-4sin3θ cos3θ=4cos3θ-3cosθ →sin3θ= ? (3sinθ-sin3θ) →cos3θ=?(3cosθ+cos3θ) xcosh-1( )= 1 x tanh-1 x dx = x tan-h1 x+ ? ln | 1-x2|+ C sin x = e jx ? e? jx cos x = a x 2 ? a 2 coth-1 x dx = x coth-1 x- ? ln | 1-x2|+ C 2 j xtanh-1( )= a x a a2 ? x2 sech-1 x dx = x sech-1 x- sin-1 x + C e jx ? e? jx csch-1 x dx = x csch-1 x+ sinh-1 x + C 2 coth-1( x )= ?a γ