DERIVATIVE AND INTEGRAL TABLES Table of Derivatives and Integrals Table of Derivatives and Integrals (Continued) Machinery's Handbook, 31st Edition
134
Derivatives
Integrals
dx d
x cos ( ) =
x sin dx ∫ =
x –cos + C
x –sin
dx d
x –ln cos | | + C
x tan ( ) = sec 2 x
x tan dx ∫ =
dx d
x ln sin | | + C
x cot ( ) =
x cot dx ∫ =
2
–csc
x
2 x ( ) sin 1 2
1 4 –--
dx d
x sec ( ) =
=
-- x + + C
2
∫
x sec
x tan
sin
xdx
dx d
1 4 --
1 2 -- x + + C
x csc ( ) =
=
2
∫
2 x ( ) sin
cos
xdx
–
x csc
x cot
1 1 x – --------- x ≠ 1, –1 2 –1 1 x – --------- x ≠ 1, –1 2
dx
b – ---------- ∫ = 2 x 2
dx d
x b -- sin –1 + C b b ( ) x 2 b 2 + – ln x
x sin –1 ( ) =
dx
dx d
x – ---------- ∫ = 2 b 2
x cos –1 ( ) =
+ C
1 1 x + ------- 2
dx + -------- ∫ = 1 b 2 x 2 b --
dx d
x b -- tan –1 + C
x tan –1 ( ) =
x b – x b + = log -------- [ ] + C
–1 1 x + ------- 2
dx d
dx – -------- ∫ = 1 b 2 x 2
x b -- tan –1 –1 2 b ---
x cot –1 ( ) =
b --
1 2 b --- x b – x b + = log -------- [ ] + C
1 x x 2 – 1 ----------- x ≠ 1, –1
dx d
dx – -------- ∫ = 1 b –-- x 2 b 2
x b -- cos –1
x sec –1 ( ) =
2 ax b + 4 ac b 2 – ------------- tan –1 [ ] + C
–1 x x 2 – 1 ----------- x ≠ 1, –1
2 4 ac b 2 – -------------
dx d
dx ax 2 bx c + + --------------- ∫ =
) =
–1
csc (
x
dx d
x sin ( ) ln ( ) =
e ax bx ( ) sin dx ∫ = a bx ( ) sin b bx ( ) – cos a 2 b 2 + -------------------------- e
ax + C
x cot
dx d dx d
a bx ( ) cos b bx ( ) + sin a 2 b 2 + -------------------------- e
x cos ( ) ln ( ) =
e ax bx ( ) cos dx ∫ =
ax + C
x –tan
1 x sin ----- dx ∫ = ln | csc x – cot x | + C 1 x cos ------ dx ∫ = ln | sec x + tan x | + C
2 2 x sin --------
x tan ( ) ln ( ) =
( ) ---
–2 2 x sin ------- ( ) ---
dx d
x cot ( ) ln ( ) =
a ax + b - -------- --
dx d
1 x + cos ----------- dx ∫ = csc x – cot x + C 1
=
ax + b
ln (
(
)
)
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