İrrasyonel fonksiyonların integralleri

Bu irrasyonel fonksiyonların integrallerini (terstürevlerini) barındıran bir listedir. Farklı fonksiyonların integrallerine ait bilgi için integral tablosu sayfasına göz atabilirsiniz.

$r = \sqrt{x^2+a^2}$ içeren integraller

$\int r \;dx = \frac{1}{2}\left(x r +a^2\,\ln\left(x+r\right)\right)$

$\int r^3 \;dx = \frac{1}{4}xr^3+\frac{1}{8}3a^2xr+\frac{3}{8}a^4\ln\left(x+r\right)$

$\int r^5 \; dx = \frac{1}{6}xr^5+\frac{5}{24}a^2xr^3+\frac{5}{16}a^4xr+\frac{5}{16}a^6\ln\left(x+r\right)$

$\int x r\;dx=\frac{r^3}{3}$

$\int x r^3\;dx=\frac{r^5}{5}$

$\int x r^{2n+1}\;dx=\frac{r^{2n+3}}{2n+3}$

$\int x^2 r\;dx= \frac{xr^3}{4}-\frac{a^2xr}{8}-\frac{a^4}{8}\ln\left(x+r\right)$

$\int x^2 r^3\;dx= \frac{xr^5}{6}-\frac{a^2xr^3}{24}-\frac{a^4xr}{16}-\frac{a^6}{16}\ln\left(x+r\right)$

$\int x^3 r \; dx = \frac{r^5}{5} - \frac{a^2 r^3}{3}$

$\int x^3 r^3 \; dx = \frac{r^7}{7}-\frac{a^2r^5}{5}$

$\int x^3 r^{2n+1} \; dx = \frac{r^{2n+5}}{2n+5} - \frac{a^3 r^{2n+3}}{2n+3}$

$\int x^4 r\;dx= \frac{x^3r^3}{6}-\frac{a^2xr^3}{8}+\frac{a^4xr}{16}+\frac{a^6}{16}\ln\left(x+r\right)$

$\int x^4 r^3\;dx= \frac{x^3r^5}{8}-\frac{a^2xr^5}{16}+\frac{a^4xr^3}{64}+\frac{3a^6xr}{128}+\frac{3a^8}{128}\ln\left(x+r\right)$

$\int x^5 r \; dx = \frac{r^7}{7} - \frac{2 a^2 r^5}{5} + \frac{a^4 r^3}{3}$

$\int x^5 r^3 \; dx = \frac{r^9}{9} - \frac{2 a^2 r^7}{7} + \frac{a^4 r^5}{5}$

$\int x^5 r^{2n+1} \; dx = \frac{r^{2n+7}}{2n+7} - \frac{2a^2r^{2n+5}}{2n+5}+\frac{a^4 r^{2n+3}}{2n+3}$

$\int\frac{r\;dx}{x} = r-a\ln\left|\frac{a+r}{x}\right| = r - a \sinh^{-1}\frac{a}{x}$

$\int\frac{r^3\;dx}{x} = \frac{r^3}{3}+a^2r-a^3\ln\left|\frac{a+r}{x}\right|$

$\int\frac{r^5\;dx}{x} = \frac{r^5}{5}+\frac{a^2r^3}{3}+a^4r-a^5\ln\left|\frac{a+r}{x}\right|$

$\int\frac{r^7\;dx}{x} = \frac{r^7}{7}+\frac{a^2r^5}{5}+\frac{a^4r^3}{3}+a^6r-a^7\ln\left|\frac{a+r}{x}\right|$

$\int\frac{dx}{r} = \sinh^{-1}\frac{x}{a} = \ln\left|x+r\right|$

$\int\frac{dx}{r^3} = \frac{x}{a^2r}$

$\int\frac{x\,dx}{r} = r$

$\int\frac{x\,dx}{r^3} = -\frac{1}{r}$

$\int\frac{x^2\;dx}{r} = \frac{x}{2}r-\frac{a^2}{2}\,\sinh^{-1}\frac{x}{a} = \frac{x}{2}r-\frac{a^2}{2}\ln\left|x+r\right|$

$\int\frac{dx}{xr} = -\frac{1}{a}\,\sinh^{-1}\frac{a}{x} = -\frac{1}{a}\ln\left|\frac{a+r}{x}\right|$

$s = \sqrt{x^2-a^2}$ içeren integraller

$(x^2>a^2)$ olduğunu varsayın. $(x^2<a^2)$ için bir sonraki bölüme bakınız.

$\int xs\;dx = \frac{1}{3}s^3$

$\int\frac{s\;dx}{x} = s - a\cos^{-1}\left|\frac{a}{x}\right|$

$\int\frac{dx}{s} = \int\frac{dx}{\sqrt{x^2-a^2}} =\ln\left|\frac{x+s}{a}\right|$

$\ln\left|\frac{x+s}{a}\right|
=\mathrm{sgn}(x)\cosh^{-1}\left|\frac{x}{a}\right|
=\frac{1}{2}\ln\left(\frac{x+s}{x-s}\right)$, burada $\cosh^{-1}\left|\frac{x}{a}\right|$'nın pozitif değeri alınmalıdır.

$\int\frac{x\;dx}{s} = s$

$\int\frac{x\;dx}{s^3} = -\frac{1}{s}$

$\int\frac{x\;dx}{s^5} = -\frac{1}{3s^3}$

$\int\frac{x\;dx}{s^7} = -\frac{1}{5s^5}$

$\int\frac{x\;dx}{s^{2n+1}} = -\frac{1}{(2n-1)s^{2n-1}}$

$\int\frac{x^{2m}\;dx}{s^{2n+1}} = -\frac{1}{2n-1}\frac{x^{2m-1}}{s^{2n-1}}+\frac{2m-1}{2n-1}\int\frac{x^{2m-2}\;dx}{s^{2n-1}}$

$\int\frac{x^2\;dx}{s} = \frac{xs}{2}+\frac{a^2}{2}\ln\left|\frac{x+s}{a}\right|$

$\int\frac{x^2\;dx}{s^3} = -\frac{x}{s}+\ln\left|\frac{x+s}{a}\right|$

$\int\frac{x^4\;dx}{s} = \frac{x^3s}{4}+\frac{3}{8}a^2xs+\frac{3}{8}a^4\ln\left|\frac{x+s}{a}\right|$

$\int\frac{x^4\;dx}{s^3} = \frac{xs}{2}-\frac{a^2x}{s}+\frac{3}{2}a^2\ln\left|\frac{x+s}{a}\right|$

$\int\frac{x^4\;dx}{s^5} = -\frac{x}{s}-\frac{1}{3}\frac{x^3}{s^3}+\ln\left|\frac{x+s}{a}\right|$

$\int\frac{x^{2m}\;dx}{s^{2n+1}} = (-1)^{n-m}\frac{1}{a^{2(n-m)}}\sum_{i=0}^{n-m-1}\frac{1}{2(m+i)+1}{n-m-1 \choose i}\frac{x^{2(m+i)+1}}{s^{2(m+i)+1}}\qquad\mbox{(}n>m\ge0\mbox{)}$

$\int\frac{dx}{s^3}=-\frac{1}{a^2}\frac{x}{s}$

$\int\frac{dx}{s^5}=\frac{1}{a^4}\left[\frac{x}{s}-\frac{1}{3}\frac{x^3}{s^3}\right]$

$\int\frac{dx}{s^7} =-\frac{1}{a^6}\left[\frac{x}{s}-\frac{2}{3}\frac{x^3}{s^3}+\frac{1}{5}\frac{x^5}{s^5}\right]$

$\int\frac{dx}{s^9} =\frac{1}{a^8}\left[\frac{x}{s}-\frac{3}{3}\frac{x^3}{s^3}+\frac{3}{5}\frac{x^5}{s^5}-\frac{1}{7}\frac{x^7}{s^7}\right]$

$\int\frac{x^2\;dx}{s^5}=-\frac{1}{a^2}\frac{x^3}{3s^3}$

$\int\frac{x^2\;dx}{s^7} = \frac{1}{a^4}\left[\frac{1}{3}\frac{x^3}{s^3}-\frac{1}{5}\frac{x^5}{s^5}\right]$

$\int\frac{x^2\;dx}{s^9} = -\frac{1}{a^6}\left[\frac{1}{3}\frac{x^3}{s^3}-\frac{2}{5}\frac{x^5}{s^5}+\frac{1}{7}\frac{x^7}{s^7}\right]$

$t = \sqrt{a^2-x^2}$ içeren integraller

$\int t \;dx = \frac{1}{2}\left(xt+a^2\arcsin\frac{x}{a}\right) \qquad\mbox{(}|x|\leq|a|\mbox{)}$

$\int xt\;dx = -\frac{1}{3} t^3 \qquad\mbox{(}|x|\leq|a|\mbox{)}$

$\int\frac{t\;dx}{x} = t-a\ln\left|\frac{a+t}{x}\right| \qquad\mbox{(}|x|\leq|a|\mbox{)}$

$\int\frac{dx}{t} = \arcsin\frac{x}{a} \qquad\mbox{(}|x|\leq|a|\mbox{)}$

$\int\frac{x^2\;dx}{t} = \frac{1}{2}\left(-xt+a^2\arcsin\frac{x}{a}\right) \qquad\mbox{(}|x|\leq|a|\mbox{)}$

$\int t\;dx = \frac{1}{2}\left(xt-\sgn x\,\cosh^{-1}\left|\frac{x}{a}\right|\right) \qquad\mbox{(for}|x|\ge|a|\mbox{)}$

$R = \sqrt{ax^2+bx+c}$ içeren integraller

$\int\frac{dx}{R} = \frac{1}{\sqrt{a}}\ln\left|2\sqrt{a}R+2ax+b\right| \qquad \mbox{(for }a>0\mbox{)}$

$\int\frac{dx}{R} = \frac{1}{\sqrt{a}}\,\sinh^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}} \qquad \mbox{(for}a>0\mbox{, }4ac-b^2>0\mbox{)}$

$\int\frac{dx}{R} = \frac{1}{\sqrt{a}}\ln|2ax+b| \quad \mbox{(for }a>0\mbox{, }4ac-b^2=0\mbox{)}$

$\int\frac{dx}{R} = -\frac{1}{\sqrt{-a}}\arcsin\frac{2ax+b}{\sqrt{b^2-4ac}} \qquad \mbox{(for}a<0\mbox{, }4ac-b^2<0\mbox{, }\left|2ax+b\right|<\sqrt{b^2-4ac}\mbox{)}$

$\int\frac{dx}{R^3} = \frac{4ax+2b}{(4ac-b^2)R}$

$\int\frac{dx}{R^5} = \frac{4ax+2b}{3(4ac-b^2)R}\left(\frac{1}{R^2}+\frac{8a}{4ac-b^2}\right)$

$\int\frac{dx}{R^{2n+1}} = \frac{2}{(2n-1)(4ac-b^2)}\left(\frac{2ax+b}{R^{2n-1}}+4a(n-1)\int\frac{dx}{R^{2n-1}}\right)$

$\int\frac{x}{R}\;dx = \frac{R}{a}-\frac{b}{2a}\int\frac{dx}{R}$

$\int\frac{x}{R^3}\;dx = -\frac{2bx+4c}{(4ac-b^2)R}$

$\int\frac{x}{R^{2n+1}}\;dx = -\frac{1}{(2n-1)aR^{2n-1}}-\frac{b}{2a}\int\frac{dx}{R^{2n+1}}$

$\int\frac{dx}{xR}=-\frac{1}{\sqrt{c}}\ln\left(\frac{2\sqrt{c}R+bx+2c}{x}\right)$

$\int\frac{dx}{xR}=-\frac{1}{\sqrt{c}}\sinh^{-1}\left(\frac{bx+2c}{|x|\sqrt{4ac-b^2}}\right)$

İNTEGRAL (Türevi verilen ifadenin eski halini bulma)

∫▒〖x^n.dx=x^(n+1)/(n+1)+c〗 ∫▒〖dx=x+c〗 ∫▒〖d(x^2+5.x-9)=x^2+5.x-9〗 ∫▒〖dp=p+c〗 ∫▒〖5.dx=5.x+c〗 ∫▒〖x.dx=x^2/2+c〗 ∫▒〖x^2.dx=x^3/3+c〗 ∫▒〖x^4.dx=x^5/5+c〗 ∫▒〖〖(x〗^2+3.x+7)dx〗=〖x/3〗^3+3.〖x/2〗^2+7.x+c ∫▒〖1/x.dx=ln⁡x 〗+c ∫▒1/x^2 .dx=-1/x+c ∫▒1/x^3 .dx=-1/(2x^2 )+c ∫▒1/x^4 .dx=-1/(3x^3 )+c ∫▒〖3.〗 ∛(x^5 ).dx=3.∫▒〖x^(5/3).dx=3.x^(5/3+1)/(5/3+1)+c〗=24/5 .∛(x^8 )+c ∫▒〖dx/∛(x^2 )=∫▒x^(- 2/3) 〗.dx=x^(-2/3+1)/(-2/3+1)+c=∛x/3+c ∫▒〖t^7.dt=t^8/8+c〗 ∫▒〖x.dt〗=x.t+c

PAYDANIN TÜREVİ PAYI VERİYORSA ∫▒〖f'(x)/f(x) .dx〗=ln|f(x)|+c ∫▒〖1/x.dx=ln⁡|x|+c〗 ∫▒〖5/(x+1).dx=5.ln|x+1| 〗+c ∫▒〖(2.x)/(x^2+9).dx=ln|x^2+9|+c〗 ∫▒〖(2.x+3)/(x^2+3x+7).dx=ln|x^2+3.x+7|+c〗 ∫▒〖(x+3)/(x^2+6.x-13).dx=ln|x^2+6.x-13|+c〗 ∫▒〖( 3/x+2/(3x+4) 〗).dx=3.ln|x|+2/3 ln|3x+4|+c ∫▒〖(〖3.e〗^x+3)/(e^x+x).dx=3.ln|e^x+x|+c〗 ∫▒〖cotx.dx=∫▒〖cosx/sinx.dx=ln|sinx|+c〗〗 ∫▒〖tgx.dx=∫▒〖sinx/cosx.dx=-ln|cosx| 〗+c〗 ∫▒(lnx+1)/(x.lnx).dx=ln|x.lnx|+c

ÜSTEL FONKSİYONLARIN İNTERALİ

   ∫▒〖a^(kx+p).dx=a^(kx+p)/(k.lna)+c〗                ∫▒〖a^x.dx=a^x/lna+c〗    
     ∫▒〖e^(kx+p).dx=e^(kx+p)/k+c〗              ∫▒〖e^x.dx=e^x+c〗    

∫▒〖5^(3x+7).dx=5^(3x+7)/(3.ln5)+c〗 ∫▒〖9^x.dx=9^x/ln9+c〗 ∫▒〖7^5x.dx=7^5x/(5.ln7)+c〗 ∫▒〖e^(4x+5).dx=e^(4x+5)/4〗+c ∫▒〖e^x.dx=e^x+c〗 ∫▒〖e^2x.dx=e^2x/2+c〗

TRİGONOMETRİK FONKSİYONLARIN İNTEGRAL

       ∫▒sin⁡〖(ax+b).dx= (-cos⁡(ax+b))/a+c〗 
       ∫▒〖cos⁡(ax+b).dx=(sin⁡(ax+b))/a+c〗
       ∫▒〖sinx.dx=-cosx+c〗
       ∫▒〖cosx.dx=sinx+c〗
       ∫▒〖dx/(cos^2 x)=∫▒〖(1+tg^2 x)〗.dx=tgx+c〗
       ∫▒〖dx/(sin^2 x)=∫▒〖(1+cot^2 x).dx=-cotx+c〗〗
       ∫▒〖dx/√(1-x^2 )=arcsinx+c_1=-arccosx+c_2 〗
       ∫▒〖dx/(1+x^2 )=arctgx+c_1=-arccotx+c_2 〗
       ∫▒〖dx/√(a^2-x^2 )=arcsin x/a+c〗
       ∫▒〖dx/(a^2+x^2 )=1/a〗.arctgx/a+c

$S = \sqrt{ax+b}$ içeren integraller

$\int \frac{dx}{x\sqrt{ax + b}}\,=\,\frac{-2}{\sqrt{b}}\tanh^{-1}{\sqrt{\frac{ax + b}{b}}}$

$\int\frac{\sqrt{ax + b}}{x}\,dx\;=\;2\left(\sqrt{ax + b} - \sqrt{b}\tanh^{-1}{\sqrt{\frac{ax + b}{b}}}\right)$

$\int\frac{x^n}{\sqrt{ax + b}}\,dx\;=\;\frac{2}{a(2n+1)} \left(x^{n}\sqrt{ax + b} - bn\int\frac{x^{n-1}}{\sqrt{ax + b}}\,dx \right)$

$\int x^n \sqrt{ax + b}\,dx \; = \; \frac{2}{2n +1}\left(x^{n+1} \sqrt{ax + b} + bx^{n} \sqrt{ax + b} - nb\int x^{n-1}\sqrt{ax + b}\,dx \right)$

Kaynakça

• 12 Şubat 2007 tarihli, İngilizce Vikipedi List of integrals of irrational functions maddesi
• Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables 1972, Dover: New York. (See chapter 3.)

Ayrıca bakınız

• İntegral tablosu