ASAL USUL RUMUS INTEGRAL

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 Identitiesarccos 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