أخي العزيز ..
اليك اليك بعضا من قوانين التكامل :
قوانين التكامل :
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
arc sin x dx = x arc sin x + (1-x2) + C
arc csc x dx = x arc cos x - (1-x2) + C
arc tan x dx = x arc tan x - (1/2) ln(1+x2) + C
Inverse Trigonometric Result
dx
________________________________________
(1 - x2)
= arc sin x + C
dx
________________________________________
x (x2 - 1)
= arc sec |x| + C
dx
________________________________________1 + x2 = arc tan x + C
Useful Identities
Arc cos x = /2 - arc sin x
(-1 <= x <= 1)
Arc csc x = /2 - arc sec x
(|x| >= 1)
Arc cot x = /2 - arc tan 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 = arc tan (sinh x) + C
tanh x dx = ln (cosh x) + C
Proof
coth x dx = ln |sinh x| + C
Proof
وفيما بعد نرسل المسائل . أخيك جورج غالي