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Selasa, 19 Oktober 2010

INTEGRAL ln(x)

ln(x) dx = x ln(x) - x + C.


1. Proof
Strategy: Use Integration by Parts.

ln(x) dx

set
u = ln(x), dv = dx
then we find
du = (1/x) dx, v = x

substitute

ln(x) dx = u dv

and use integration by parts

= uv - v du

substitute u=ln(x), v=x, and du=(1/x)dx

= ln(x) x - x (1/x) dx
= ln(x) x - dx
= ln(x) x - x + C

= x ln(x) - x + C.

Power of x. xn dx = xn+1 (n+1)-1 + C
(n -1) Proof x-1 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

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