Power of x. xn dx = x(n+1) / (n+1) + C
(n -1) Proof 1/x dx = ln
x
+ C
Exponential / Logarithmic ex dx = ex + C
Proof bx dx = bx / ln(b) + C
Proof, Tip!
ln(x) dx = x ln(x) - x + C
Proof
Trigonometric sin x dx = -cos x + C
Proof csc x dx = - ln
CSC x + cot x
+ C
Proof
COs x dx = sin x + C
Proof sec x dx = ln
sec x + tan x
+ C
Proof
tan x dx = -ln
COs x
+ C
Proof cot x dx = ln
sin x
+ C
Proof
Trigonometric Result COs x dx = sin x + C
Proof CSC x cot x dx = - CSC x + C
Proof
sin x dx = COs x + C
Proof sec x tan x dx = sec x + C
Proof
sec2 x dx = tan x + C
Proof csc2 x dx = - cot x + C
Proof
Inverse Trigonometric arcsin x dx = x arcsin x + (1-x2) + C
arccsc x dx = x arccos x - (1-x2) + C
arctan x dx = x arctan x - (1/2) ln(1+x2) + C
Inverse Trigonometric Result
dx
(1 - x2) = arcsin x + C
dx
x (x2 - 1) = arcsec
x
+ C
dx
1 + x2 = arctan x + C
Useful Identities
arccos x = /2 - arcsin x
(-1 <= x <= 1)
arccsc x = /2 - arcsec x
(
x
>= 1)
arccot x = /2 - arctan x
(for all x)
Hyperbolic sinh x dx = cosh x + C
Proof csch x dx = ln
tanh(x/2)
+ C
Proof
cosh x dx = sinh x + C
Proof sech x dx = arctan (sinh x) + C
tanh x dx = ln (cosh x) + C
Proof coth x dx = ln
sinh x
+ C
Proof
