rumus integral lengkap

integral e^x

Discussion of(integral) ex dx = ex + C.Since we know the derivative: (d/dx) ex = ex,
we can use the Fundamental Theorem of calculus:
(integral) ex dx = (integral) (d/dx) (ex) dx = ex + C
See also the proof that (d/dx) ex = ex.

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 Identitiesarccos 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

Formal Integral Definition:
(integral)(a to b) f(x) dx = lim (d -> 0) (sum) (k=1..n) f(X(k)) (x(k) – x(k-1)) when…

a = x0 < x1 < x2 < … < xn = b

d = max (x1-x0, x2-x1, … , xn – x(n-1))

x(k-1) <= X(k) <= x(k)     k = 1, 2, … , n

(integral)(a to b) F ‘(x) dx = F(b) – F(a) (Fundamental Theorem for integrals of derivatives)


(integral)a f(x) dx = a(integral) f(x) dx (if a is constant)

(integral)f(x) + g(x) dx = (integral)f(x) dx + (integral)g(x) dx

(integral)(a to b) f(x) dx = (integral)f(x) dx | (a b)

(integral)(a to b) f(x) dx + (integral)(b to c) f(x) dx = (integral)(a to c) f(x) dx

(integral)f(u) du/dx dx = (integral)f(u) du (integration by substitution)

Some of these functions I have seen defined under both intervals (0 to x) and (x to inf). In that case, both variant definitions are listed.
gamma = Euler’s constant = 0.5772156649…

(x) = Gamma(x) = (integral)(0 to inf)t^(x-1) e^(-t)dt (Gamma function)
B(x,y) = (integral)(0 to 1)t^(x-1) (1-t)^(y-1)DT
(Beta function)
Ei(x) = (integral)(x to inf)e^(-t)/t DT (exponential integral) or it’s variant, NONEQUIVALENT form:

Ei(x) = + ln(x) + (integral)(0 to x)(e^t – 1)/t DT = gamma + ln(x) + (sum)(n=1..inf)x^n/(n*n!)

li(x) = (integral)(2 to x)1/ln(t) DT (logarithmic integral)
Si(x) = (integral)(x to inf)sin(t)/t DT (sine integral) or it’s variant, NONEQUIVALENT form:

Si(x) = (integral)(0 to<br />
x)” align=”MIDDLE” />sin(t)/t DT = PI/2 – <img src=(x to inf)sin(t)/t DT


Ci(x) = (integral)(x to inf)cos(t)/t DT (cosine integral) or it’s variant, NONEQUIVALENT form:

CI(x) = - (integral)(x<br />
to inf)” align=”MIDDLE” />COs(t)/t DT = gamma + ln(x) + <img src=(0 to<br />
x)” align=”MIDDLE” /> (COs(t) – 1) / t DT (cosine integral)</p></blockquote>
<p><strong><br />
Chi(x) = gamma + ln(x) + <img src=(0 to<br />
x)” align=”MIDDLE” />(cosh(t)-1)/t DT <span style=(hyperbolic cosine integral)
Shi(x) = (integral)(0 to x)sinh(t)/t DT (hyperbolic sine integral)
Erf(x) = 2/PI^(1/2)(integral)(0 to<br />
x)” align=”MIDDLE” />e<sup>^(-t^2)</sup> DT = 2/<img src=PI (sum)(n=0..inf) (-1)^n x^(2n+1) / ( n! (2n+1) ) (error function)
FresnelC(x) = (integral)(0 to x)COs(PI/2 t^2) DT
FresnelS(x) = (integral)(0 to x)sin(PI/2 t^2) DT
dilog(x) = (integral)(1 to x)ln(t)/(1-t) DT
Psi(x) = (d/dx)ln(Gamma(x))
Psi(n,x) = nth derivative of Psi(x)
W(x) = inverse of x*e^x
L sub n (x) = (e^x/n!)( x^n e^(-x) ) (n) (laguerre polynomial degree n. (n) meaning nth derivative)
Zeta(s) = (sum)(n=1..inf) 1/n^s

Dirichlet’s beta function B(x) = (sum)(n=0..inf) (-1)^n / (2n+1)^x

math.com

About these ads

Tinggalkan Balasan

Isikan data di bawah atau klik salah satu ikon untuk log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Logout / Ubah )

Twitter picture

You are commenting using your Twitter account. Logout / Ubah )

Facebook photo

You are commenting using your Facebook account. Logout / Ubah )

Google+ photo

You are commenting using your Google+ account. Logout / Ubah )

Connecting to %s