Table of Integrals

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