Daftar integral dari fungsi irrasional


Dari Wikipedia Indonesia, artikel bebas.

 

Integral melibatkan r = sqrt{x^2+a^2}

int r ;dx = frac{1}{2}left(x r +a^2,lnleft(x+rright)right)
int r^3 ;dx = frac{1}{4}xr^3+frac{1}{8}3a^2xr+frac{3}{8}a^4lnleft(x+rright)
int r^5 ; dx = frac{1}{6}xr^5+frac{5}{24}a^2xr^3+frac{5}{16}a^4xr+frac{5}{16}a^6lnleft(x+rright)
int x r;dx=frac{r^3}{3}
int x r^3;dx=frac{r^5}{5}
int x r^{2n+1};dx=frac{r^{2n+3}}{2n+3}
int x^2 r;dx= frac{xr^3}{4}-frac{a^2xr}{8}-frac{a^4}{8}lnleft(x+rright)
int x^2 r^3;dx= frac{xr^5}{6}-frac{a^2xr^3}{24}-frac{a^4xr}{16}-frac{a^6}{16}lnleft(x+rright)
int x^3 r ; dx = frac{r^5}{5} - frac{a^2 r^3}{3}
int x^3 r^3 ; dx = frac{r^7}{7}-frac{a^2r^5}{5}
int x^3 r^{2n+1} ; dx = frac{r^{2n+5}}{2n+5} - frac{a^3 r^{2n+3}}{2n+3}
int x^4 r;dx= frac{x^3r^3}{6}-frac{a^2xr^3}{8}+frac{a^4xr}{16}+frac{a^6}{16}lnleft(x+rright)
int x^4 r^3;dx= frac{x^3r^5}{8}-frac{a^2xr^5}{16}+frac{a^4xr^3}{64}+frac{3a^6xr}{128}+frac{3a^8}{128}lnleft(x+rright)
int x^5 r ; dx = frac{r^7}{7} - frac{2 a^2 r^5}{5} + frac{a^4 r^3}{3}
int x^5 r^3 ; dx = frac{r^9}{9} - frac{2 a^2 r^7}{7} + frac{a^4 r^5}{5}
int x^5 r^{2n+1} ; dx = frac{r^{2n+7}}{2n+7} - frac{2a^2r^{2n+5}}{2n+5}+frac{a^4 r^{2n+3}}{2n+3}
intfrac{r;dx}{x} = r-alnleft|frac{a+r}{x}right| = r - a sinh^{-1}frac{a}{x}
intfrac{r^3;dx}{x} = frac{r^3}{3}+a^2r-a^3lnleft|frac{a+r}{x}right|
intfrac{r^5;dx}{x} = frac{r^5}{5}+frac{a^2r^3}{3}+a^4r-a^5lnleft|frac{a+r}{x}right|
intfrac{r^7;dx}{x} = frac{r^7}{7}+frac{a^2r^5}{5}+frac{a^4r^3}{3}+a^6r-a^7lnleft|frac{a+r}{x}right|
intfrac{dx}{r} = sinh^{-1}frac{x}{a} = lnleft|x+rright|
intfrac{dx}{r^3} = frac{x}{a^2r}
intfrac{x,dx}{r} = r
intfrac{x,dx}{r^3} = -frac{1}{r}
intfrac{x^2;dx}{r} = frac{x}{2}r-frac{a^2}{2},sinh^{-1}frac{x}{a} = frac{x}{2}r-frac{a^2}{2}lnleft|x+rright|
intfrac{dx}{xr} = -frac{1}{a},sinh^{-1}frac{a}{x} = -frac{1}{a}lnleft|frac{a+r}{x}right|

 

 

Integral melibatkan s = sqrt{x^2-a^2}

 

Anggap (x2 > a2), untuk (x2 < a2), perhatikan bagian berikutnya:

int xs;dx = frac{1}{3}s^3
intfrac{s;dx}{x} = s - acos^{-1}left|frac{a}{x}right|
intfrac{dx}{s} = intfrac{dx}{sqrt{x^2-a^2}} =lnleft|frac{x+s}{a}right|

Perhatikan bahwa lnleft|frac{x+s}{a}right|
=mathrm{sgn}(x)cosh^{-1}left|frac{x}{a}right|
=frac{1}{2}lnleft(frac{x+s}{x-s}right), dimana nilai positif dari cosh^{-1}left|frac{x}{a}right| lah yang diambil.

intfrac{x;dx}{s} = s
intfrac{x;dx}{s^3} = -frac{1}{s}
intfrac{x;dx}{s^5} = -frac{1}{3s^3}
intfrac{x;dx}{s^7} = -frac{1}{5s^5}
intfrac{x;dx}{s^{2n+1}} = -frac{1}{(2n-1)s^{2n-1}}
intfrac{x^{2m};dx}{s^{2n+1}}
= -frac{1}{2n-1}frac{x^{2m-1}}{s^{2n-1}}+frac{2m-1}{2n-1}intfrac{x^{2m-2};dx}{s^{2n-1}}
intfrac{x^2;dx}{s}
= frac{xs}{2}+frac{a^2}{2}lnleft|frac{x+s}{a}right|
intfrac{x^2;dx}{s^3}
= -frac{x}{s}+lnleft|frac{x+s}{a}right|
intfrac{x^4;dx}{s}
= frac{x^3s}{4}+frac{3}{8}a^2xs+frac{3}{8}a^4lnleft|frac{x+s}{a}right|
intfrac{x^4;dx}{s^3}
= frac{xs}{2}-frac{a^2x}{s}+frac{3}{2}a^2lnleft|frac{x+s}{a}right|
intfrac{x^4;dx}{s^5}
= -frac{x}{s}-frac{1}{3}frac{x^3}{s^3}+lnleft|frac{x+s}{a}right|
mge0mbox{)}” />
intfrac{dx}{s^3}=-frac{1}{a^2}frac{x}{s}
intfrac{dx}{s^5}=frac{1}{a^4}left[frac{x}{s}-frac{1}{3}frac{x^3}{s^3}right]
intfrac{dx}{s^7}
=-frac{1}{a^6}left[frac{x}{s}-frac{2}{3}frac{x^3}{s^3}+frac{1}{5}frac{x^5}{s^5}right]
intfrac{dx}{s^9}
=frac{1}{a^8}left[frac{x}{s}-frac{3}{3}frac{x^3}{s^3}+frac{3}{5}frac{x^5}{s^5}-frac{1}{7}frac{x^7}{s^7}right]
intfrac{x^2;dx}{s^5}=-frac{1}{a^2}frac{x^3}{3s^3}
intfrac{x^2;dx}{s^7}
= frac{1}{a^4}left[frac{1}{3}frac{x^3}{s^3}-frac{1}{5}frac{x^5}{s^5}right]
intfrac{x^2;dx}{s^9}
= -frac{1}{a^6}left[frac{1}{3}frac{x^3}{s^3}-frac{2}{5}frac{x^5}{s^5}+frac{1}{7}frac{x^7}{s^7}right]

 

Integral melibatkan t = sqrt{a^2-x^2}

 

int t ;dx = frac{1}{2}left(xt+a^2arcsinfrac{x}{a}right) qquadmbox{(}|x|leq|a|mbox{)}
int xt;dx = -frac{1}{3} t^3 qquadmbox{(}|x|leq|a|mbox{)}
intfrac{t;dx}{x} = t-alnleft|frac{a+t}{x}right| qquadmbox{(}|x|leq|a|mbox{)}
intfrac{dx}{t} = arcsinfrac{x}{a} qquadmbox{(}|x|leq|a|mbox{)}
intfrac{x^2;dx}{t} = frac{1}{2}left(-xt+a^2arcsinfrac{x}{a}right) qquadmbox{(}|x|leq|a|mbox{)}
int t;dx = frac{1}{2}left(xt-sgn x,cosh^{-1}left|frac{x}{a}right|right) qquadmbox{(untuk }|x|ge|a|mbox{)}

 

 

Integral melibatkan R = sqrt{ax^2+bx+c}

 

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0mbox{, }4ac-b^2>0mbox{)}” />
0mbox{, }4ac-b^2=0mbox{)}” />
<img src="http://upload.wikimedia.org/math/0/6/0/0600ebc168bcda07ca9f11c478dd345b.png&quot; alt="intfrac{dx}{R} = -frac{1}{sqrt{-a}}arcsinfrac{2ax+b}{sqrt{b^2-4ac}} qquad mbox{(untuk }a<0mbox{, }4ac-b^2<0mbox{, }left|2ax+bright|
intfrac{dx}{R^3} = frac{4ax+2b}{(4ac-b^2)R}
intfrac{dx}{R^5} = frac{4ax+2b}{3(4ac-b^2)R}left(frac{1}{R^2}+frac{8a}{4ac-b^2}right)
intfrac{dx}{R^{2n+1}} = frac{2}{(2n-1)(4ac-b^2)}left(frac{2ax+b}{R^{2n-1}}+4a(n-1)intfrac{dx}{R^{2n-1}}right)
intfrac{x}{R};dx = frac{R}{a}-frac{b}{2a}intfrac{dx}{R}
intfrac{x}{R^3};dx = -frac{2bx+4c}{(4ac-b^2)R}
intfrac{x}{R^{2n+1}};dx = -frac{1}{(2n-1)aR^{2n-1}}-frac{b}{2a}intfrac{dx}{R^{2n+1}}
intfrac{dx}{xR}=-frac{1}{sqrt{c}}lnleft(frac{2sqrt{c}R+bx+2c}{x}right)
intfrac{dx}{xR}=-frac{1}{sqrt{c}}sinh^{-1}left(frac{bx+2c}{|x|sqrt{4ac-b^2}}right)

 

 

Integral melibatkan S = sqrt{ax+b}

int frac{dx}{xsqrt{ax + b}},=,frac{-2}{sqrt{b}}tanh^{-1}{sqrt{frac{ax + b}{b}}}
intfrac{sqrt{ax + b}}{x},dx;=;2left(sqrt{ax + b} - sqrt{b}tanh^{-1}{sqrt{frac{ax + b}{b}}}right)
intfrac{x^n}{sqrt{ax + b}},dx;=;frac{2}{a(2n+1)}
left(x^{n}sqrt{ax + b} - bnintfrac{x^{n-1}}{sqrt{ax + b}},dx right)
int x^n sqrt{ax + b},dx ; = ; frac{2}{2n +1}left(x^{n+1} sqrt{ax + b} + bx^{n} sqrt{ax + b} - nbint x^{n-1}sqrt{ax + b},dx right)

 

Budi Santoso

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