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

 

To solve a more complicated integral, see The Integrator at http://integrals.wolfram.com/

 

 
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