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Power of x.

(integral)xn dx = x(n+1) / (n+1) + C 
(n  -1)  Proof
(integral)1/x dx = ln|x| + C

Exponential / Logarithmic

(integral)ex dx = ex + C  
Proof 
(integral)bx dx = bx / ln(b) + C  
Proof, Tip!
(integral)ln(x) dx = x ln(x) - x + C  
Proof

Trigonometric

(integral)sin x dx = -cos x + C  
Proof
(integral)csc x dx = - ln|CSC x + cot x| + C  
Proof
(integral)COs x dx = sin x + C  
Proof
(integral)sec x dx = ln|sec x + tan x| + C  
Proof
(integral)tan x dx = -ln|COs x| + C  
Proof
(integral)cot x dx = ln|sin x| + C  
Proof

Trigonometric Result

(integral)COs x dx = sin x + C   
Proof
(integral)CSC x cot x dx = - CSC x + C   
Proof
(integral)sin x dx = COs x + C   
Proof
(integral)sec x tan x dx = sec x + C   
Proof
(integral)sec2 x dx = tan x + C   
Proof
(integral)csc2 x dx = - cot x + C   
Proof

Inverse Trigonometric

(integral)arcsin x dx = x arcsin x + sqrt(1-x2) + C
(integral)arccsc x dx = x arccos x - sqrt(1-x2) + C
(integral)arctan x dx = x arctan x - (1/2) ln(1+x2) + C

Inverse Trigonometric Result
 

(integral)  dx 
sqrt(1 - x2)
 = arcsin x + C
 
(integral)  dx 
sqrt(x2 - 1)
 = arcsec|x| + C
 
(integral)  dx 
1 + x2
 = arctan x + C
 
 
Useful Identities

arccos x = pi/2 - arcsin x 
(-1 <= x <= 1) 

arccsc x = pi/2 - arcsec x 
(|x| >= 1) 

arccot x = pi/2 - arctan x 
(for all x)

 

Hyperbolic

(integral)sinh x dx = cosh x + C   
Proof
(integral)csch x dx = ln |tanh(x/2)| + C   
Proof
(integral)cosh x dx = sinh x + C   
Proof
(integral)sech x dx = arctan (sinh x) + C
(integral)tanh x dx = ln (cosh x) + C   
Proof
(integral)coth x dx = ln |sinh x| + C  
Proof

 

http://www.math.com/tables/integrals/tableof.htm

http://integrals.wolfram.com/