Integration and Differentiation

Differentiation Formulas Integration Formulas


Z
d
k=0 (1) dx = x + C (1)
dx
d
[f (x) ± g(x)] = f 0 (x) ± g 0 (x) xn+1
Z
(2)
dx xn dx = +C (2)
n+1
d
[k · f (x)] = k · f 0 (x) (3) Z
dx
dx = ln |x| + C (3)
d x
[f (x)g(x)] = f (x)g 0 (x) + g(x)f 0 (x) (4)
dx Z

d f (x)

g(x)f 0 (x) − f (x)g 0 (x) ex dx = ex + C (4)
= 2 (5)
dx g(x) [g(x)] Z
1 x
d ax dx = a +C (5)
f (g(x)) = f 0 (g(x)) · g 0 (x) (6) ln a
dx Z
d n ln x dx = x ln x − x + C (6)
x = nxn−1 (7)
dx
d
Z
sin x = cos x (8) sin x dx = − cos x + C (7)
dx
d Z
cos x = − sin x (9)
dx cos x dx = sin x + C (8)
d
tan x = sec2 x (10) Z
dx tan x dx = − ln | cos x| + C (9)
d
cot x = − csc2 x (11)
dx Z
d cot x dx = ln | sin x| + C (10)
sec x = sec x tan x (12)
dx Z
d sec x dx = ln | sec x + tan x| + C (11)
csc x = − csc x cot x (13)
dx
d x
Z
e = ex (14) csc x dx = − ln | csc x + cot x| + C (12)
dx
d x
a = ax ln a
Z
(15)
dx sec2 x dx = tan x + C (13)
d 1
ln |x| = (16) Z
dx x csc2 x dx = − cot x + C (14)
d 1
sin−1 x = √ (17)
dx 1 − x2 Z
sec x tan x dx = sec x + C (15)
d −1
cos−1 x = √ (18)
dx 1 − x2 Z
d 1 csc x cot x dx = − csc x + C (16)
tan−1 x = 2 (19)
dx x +1 Z
dx x
d −1 √ = sin−1 + C (17)
cot−1 x = 2 (20) a2 − x2 a
dx x +1
Z
d 1 dx 1 x
sec−1 x = √ (21) = tan−1 + C (18)
dx |x| x2 − 1 a2 + x2 a a
d −1 Z
dx 1 |x|
csc−1 x = √ (22) √ = sec−1 +C (19)
dx |x| x2 − 1 2
x x −a2 a a