kamiran

Penyepaduan ialah operasi terbalik terbitan.

Kamiran bagi suatu fungsi ialah luas di bawah graf fungsi tersebut.

Takrif Kamiran Tak Tentu

Bila dF(x)/dx = f(x) => kamiran(f(x)*dx) = F(x) + c

Sifat Kamiran Tak Tentu

kamiran(f(x)+g(x))*dx = kamiran(f(x)*dx) + kamiran(g(x)*dx)

kamiran(a*f(x)*dx) = a*kamiran(f(x)*dx)

kamiran(f(a*x)*dx) = 1/a * F(a*x)+c

kamiran(f(x+b)*dx) = F(x+b)+c

kamiran(f(a*x+b)*dx) = 1/a * F(a*x+b) + c

kamiran(df(x)/dx * dx) = f(x)

Perubahan Pembolehubah Integrasi

Bila danx = g(t)dx = g'(t)*dt

kamiran(f(x)*dx) = kamiran(f(g(t))*g'(t)*dt)

Integrasi Mengikut Bahagian

kamiran(f(x)*g'(x)*dx) = f(x)*g(x) - kamiran(f'(x)*g(x)*dx)

Jadual Kamiran

kamiran(f(x)*dx = F(x) + c

kamiran(a*dx) = a*x+c

kamiran(x^n*dx) = 1/(a+1) * x^(a+1) + c , apabila a<>-1

kamiran(1/x*dx) = ln(abs(x)) + c

kamiran(e^x*dx) = e^x + c

kamiran(a^x*dx) = a^x / ln(x) + c

kamiran(ln(x)*dx) = x*ln(x) - x + c

kamiran(sin(x)*dx) = -cos(x) + c

kamiran(cos(x)*dx) = sin(x) + c

kamiran(tan(x)*dx) = -ln(abs(cos(x))) + c

kamiran(arcsin(x)*dx) = x*arcsin(x) + sqrt(1-x^2) + c

integral(arccos(x)*dx) = x*arccos(x) - sqrt(1-x^2) + c

kamiran(arctan(x)*dx) = x*arctan(x) - 1/2*ln(1+x^2) + c

kamiran(dx/(ax+b)) = 1/a*ln(abs(a*x+b)) + c

kamiran(1/sqrt(a^2-x^2)*dx) = arcsin(x/a) + c

kamiran(1/sqrt(x^2 +- a^2)*dx) = ln(abs(x + sqrt(x^2 +- a^2)) + c

kamiran(x*sqrt(x^2-a^2)*dx) = 1/(a*arccos(x/a)) + c

kamiran(1/(a^2+x^2)*dx) = 1/a*arctan(x/a) + c

kamiran(1/(a^2-x^2)*dx) = 1/2a*ln(abs(((a+x)/(ax))) + c

kamiran(sinh(x)*dx) = cosh(x) + c

integral(cosh(x)*dx) = sinh(x) + c

integral(tanh(x)*dx) = ln(cosh(x)) + c

 

Definisi Kamiran Pasti

kamiran(a..b, f(x)*dx) = lim(n->inf, jumlah(i=1..n, f(z(i))*dx(i)))  

Bilax0=a, xn=b

dx(k) = x(k) - x(k-1)

x(k-1) <= z(k) <=x(k)

Pengiraan Kamiran Pasti

bila ,

 dF(x)/dx = f(x)  dan

kamiran(a..b, f(x)*dx) = F(b) - F(a)  

Sifat Kamiran Pasti

kamiran(a..b, (f(x)+g(x))*dx) = kamiran(a..b, f(x)*dx) + kamiran(a..b, g(x)*dx )

kamiran(a..b, c*f(x)*dx) = c*integral(a..b, f(x)*dx)

kamiran(a..b, f(x)*dx) = - kamiran(b..a, f(x)*dx)

kamiran(a..b, f(x)*dx) = kamiran(a..c, f(x)*dx) + kamiran(c..b, f(x)*dx)

abs( integral(a..b, f(x)*dx) ) <= integral(a..b, abs(f(x))*dx)

min(f(x))*(ba) <= integral(a..b, f(x)*dx) <= max(f(x))*(ba) bilax ahli [a,b]

Perubahan Pembolehubah Integrasi

bila , , ,x = g(t)dx = g'(t)*dtg(alfa) = ag(beta) = b

kamiran(a..b, f(x)*dx) = kamiran(alfa..beta, f(g(t))*g'(t)*dt)

Integrasi Mengikut Bahagian

kamiran(a..b, f(x)*g'(x)*dx) = kamiran(a..b, f(x)*g(x)*dx) - kamiran(a..b, f' (x)*g(x)*dx)

Teorem nilai min

Apabilaf (x ) selanjar ada titik jadi c ialah ahli [a,b]

kamiran(a..b, f(x)*dx) = f(c)*(ba)   

Penghampiran Trapezoid Kamiran Pasti

kamiran(a..b, f(x)*dx) ~ (ba)/n * (f(x(0))/2 + f(x(1)) + f(x(2)) +.. .+ f(x(n-1)) + f(x(n))/2)

Fungsi Gamma

gamma(x) = kamiran(0..inf, t^(x-1)*e^(-t)*dt

Fungsi Gamma adalah menumpu untukx> 0.

Sifat Fungsi Gamma

G(x+1) = xG(x)

G(n+1) = n! , when nis member ofℕ (positive integer).

Fungsi Beta

B(x,y) = kamiran(0..1, t^(n-1)*(1-t)^(y-1)*dt

Fungsi Beta dan Hubungan Fungsi Gamma

B(x,y) = Gamma(x)*Gamma(y)/Gamma(x+y)

 

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