تاریخچه ی:
جدول انتگرال توابع گنگ
تفاوت با نگارش: 3
| | ||V{maketoc}|| | | ||V{maketoc}|| |
| | ^@#16: | | ^@#16: |
| | !جدول انتگرال توابع گنگ | | !جدول انتگرال توابع گنگ |
| | @@{TEX()} {\int \frac{dx}{\sqrt{a^2-x^2}}=arcsin \frac{x}{a}+C} {TEX}@@ | | @@{TEX()} {\int \frac{dx}{\sqrt{a^2-x^2}}=arcsin \frac{x}{a}+C} {TEX}@@ |
| | @@{TEX()} {\int \frac{-dx}{\sqrt{a^2-x^2}}=arcos \frac{x}{a} +C} {TEX}@@ | | @@{TEX()} {\int \frac{-dx}{\sqrt{a^2-x^2}}=arcos \frac{x}{a} +C} {TEX}@@ |
| | @@{TEX()} {\int \frac{dx}{x\sqrt{x^2-a^2}}=\frac{1}{a} arcsec \frac{|x|}{a}+C} {TEX}@@ | | @@{TEX()} {\int \frac{dx}{x\sqrt{x^2-a^2}}=\frac{1}{a} arcsec \frac{|x|}{a}+C} {TEX}@@ |
| | --- | | --- |
| | @@||{TEX()} {r=\sqrt{x^2+a^2}} {TEX}||@@ | | @@||{TEX()} {r=\sqrt{x^2+a^2}} {TEX}||@@ |
| | @@{TEX()} {\int rdx=\frac{1}{2}\Big ( xr+a^2 ln \Big( \frac{x+r}{a} \Big) \Big)} {TEX}@@ | | @@{TEX()} {\int rdx=\frac{1}{2}\Big ( xr+a^2 ln \Big( \frac{x+r}{a} \Big) \Big)} {TEX}@@ |
| | @@{TEX()} {\int r^3dx=\frac{1}{4}xr^3+\frac{1}{8}3a^2xr+\frac{3}{8}a^4 ln\Big( \frac{x+r}{a} \Big)} {TEX}@@ | | @@{TEX()} {\int r^3dx=\frac{1}{4}xr^3+\frac{1}{8}3a^2xr+\frac{3}{8}a^4 ln\Big( \frac{x+r}{a} \Big)} {TEX}@@ |
| | @@{TEX()} {\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 \Big( \frac{x+r}{a} \Big)} {TEX}@@ | | @@{TEX()} {\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 \Big( \frac{x+r}{a} \Big)} {TEX}@@ |
| | @@{TEX()} {\int xr dx=\frac{r^3}{3}} {TEX}@@ | | @@{TEX()} {\int xr dx=\frac{r^3}{3}} {TEX}@@ |
| | @@{TEX()} {\int xr^3 dx =\frac{r^5}{5}} {TEX}@@ | | @@{TEX()} {\int xr^3 dx =\frac{r^5}{5}} {TEX}@@ |
| | @@{TEX()} {\int xr^{2n+1}dx=\frac{r^{2n+3}}{2n+3}} {TEX}@@ | | @@{TEX()} {\int xr^{2n+1}dx=\frac{r^{2n+3}}{2n+3}} {TEX}@@ |
| | @@{TEX()} {\int x^2r dx=\frac{xr^3}{4}-\frac{a^2xr}{8}-\frac{a^4}{8} ln \Big(\frac{x+r}{a} \Big)} {TEX}@@ | | @@{TEX()} {\int x^2r dx=\frac{xr^3}{4}-\frac{a^2xr}{8}-\frac{a^4}{8} ln \Big(\frac{x+r}{a} \Big)} {TEX}@@ |
| | @@{TEX()} {\int x^2r^3 dx=\frac{xr^5}{6}-\frac{a^2xr^3}{24}-\frac{a^4xr}{16}-\frac{a^6}{16} ln \Big(\frac{x+r}{a} \Big)} {TEX}@@ | | @@{TEX()} {\int x^2r^3 dx=\frac{xr^5}{6}-\frac{a^2xr^3}{24}-\frac{a^4xr}{16}-\frac{a^6}{16} ln \Big(\frac{x+r}{a} \Big)} {TEX}@@ |
| | @@{TEX()} {\int x^3 rdx =\frac{r^5}{5}-\frac{a^2r^3}{3}} {TEX}@@ | | @@{TEX()} {\int x^3 rdx =\frac{r^5}{5}-\frac{a^2r^3}{3}} {TEX}@@ |
| | @@{TEX()} {\int x^3 r^3 dx=\frac{r^7}{7}-\frac{a^2r^5}{5}} {TEX}@@ | | @@{TEX()} {\int x^3 r^3 dx=\frac{r^7}{7}-\frac{a^2r^5}{5}} {TEX}@@ |
| | @@{TEX()} {\int x^3r^{2n+1}dx=\frac{r^{2n+5}}{2n+5}-\frac{a^3r^{2n+3}}{2n+3}} {TEX}@@ | | @@{TEX()} {\int x^3r^{2n+1}dx=\frac{r^{2n+5}}{2n+5}-\frac{a^3r^{2n+3}}{2n+3}} {TEX}@@ |
| | @@{TEX()} {\int x^4r dx=\frac{x^3r^3}{6}-\frac{a^2xr^3}{8}+\frac{a^4xr}{16}+\frac{a^6}{16} ln \Big( \frac{x+r}{a} \Big)} {TEX}@@ | | @@{TEX()} {\int x^4r dx=\frac{x^3r^3}{6}-\frac{a^2xr^3}{8}+\frac{a^4xr}{16}+\frac{a^6}{16} ln \Big( \frac{x+r}{a} \Big)} {TEX}@@ |
| | @@{TEX()} {\int x^4r^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 \Big( \frac{x+r}{a} \Big)} {TEX}@@ | | @@{TEX()} {\int x^4r^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 \Big( \frac{x+r}{a} \Big)} {TEX}@@ |
| | @@{TEX()} {\int x^5r dx=\frac{r^7}{7}-\frac{2a^2r^5}{5}+\frac{a^4r^3}{3}} {TEX}@@ | | @@{TEX()} {\int x^5r dx=\frac{r^7}{7}-\frac{2a^2r^5}{5}+\frac{a^4r^3}{3}} {TEX}@@ |
| | @@{TEX()} {\int x^5r^3 dx=\frac{r^9}{9}-\frac{2a^2r^7}{7}+\frac{a^4r^5}{5}} {TEX}@@ | | @@{TEX()} {\int x^5r^3 dx=\frac{r^9}{9}-\frac{2a^2r^7}{7}+\frac{a^4r^5}{5}} {TEX}@@ |
| | @@{TEX()} {\int x^5r^{2n+1} dx=\frac{r^{2n+7}}{2n+7}-\frac{2a^2r^{2n+5}}{2n+5}+\frac{a^4r^{2n+3}}{2n+3}} {TEX}@@ | | @@{TEX()} {\int x^5r^{2n+1} dx=\frac{r^{2n+7}}{2n+7}-\frac{2a^2r^{2n+5}}{2n+5}+\frac{a^4r^{2n+3}}{2n+3}} {TEX}@@ |
| | @@{TEX()} {\int \frac{r \ dx}{x}=r-aln \Big| \frac{a+r}{x} \Big| =r-asinh^{-1} \frac{a}{x}} {TEX}@@ | | @@{TEX()} {\int \frac{r \ dx}{x}=r-aln \Big| \frac{a+r}{x} \Big| =r-asinh^{-1} \frac{a}{x}} {TEX}@@ |
| | @@{TEX()} {\int \frac{r^3 dx}{x}=\frac{r^3}{3}+a^2r –a^3 ln \Big| \frac{a+r}{x} \Big| } {TEX}@@ | | @@{TEX()} {\int \frac{r^3 dx}{x}=\frac{r^3}{3}+a^2r –a^3 ln \Big| \frac{a+r}{x} \Big| } {TEX}@@ |
| | @@{TEX()} {\int \frac{r^5 dx}{x}=\frac{r^5}{5}+\frac{a^2r^3}{3}+a^4r-a^5 ln \Big|\frac{a+r}{x }\Big|} {TEX}@@ | | @@{TEX()} {\int \frac{r^5 dx}{x}=\frac{r^5}{5}+\frac{a^2r^3}{3}+a^4r-a^5 ln \Big|\frac{a+r}{x }\Big|} {TEX}@@ |
| | @@{TEX()} {\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 \Big| \frac{a+r}{x}\Big|} {TEX}@@ | | @@{TEX()} {\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 \Big| \frac{a+r}{x}\Big|} {TEX}@@ |
| | @@{TEX()} {\int \frac{dx}{r}=sinh^{-1} \frac{x}{a}=ln \Big|x+r \Big|} {TEX}@@ | | @@{TEX()} {\int \frac{dx}{r}=sinh^{-1} \frac{x}{a}=ln \Big|x+r \Big|} {TEX}@@ |
| | @@{TEX()} {\int \frac{x \ dx}{r}=r} {TEX}@@ | | @@{TEX()} {\int \frac{x \ dx}{r}=r} {TEX}@@ |
| | @@{TEX()} {\int \frac{x^2dx}{r}=\frac{x}{2}r -\frac{a^2}{2}sinh^{-1} \frac{x}{a}= \frac{x}{2} r -\frac{a^2}{2} ln |x+r|} {TEX}@@ | | @@{TEX()} {\int \frac{x^2dx}{r}=\frac{x}{2}r -\frac{a^2}{2}sinh^{-1} \frac{x}{a}= \frac{x}{2} r -\frac{a^2}{2} ln |x+r|} {TEX}@@ |
| | @@{TEX()} {\int \frac{dx}{xr}=-\frac{1}{a} sinh^{-1} \frac{a}{x} =-\frac{1}{a} ln \Big|\frac{a+r}{x} \Big|} {TEX}@@ | | @@{TEX()} {\int \frac{dx}{xr}=-\frac{1}{a} sinh^{-1} \frac{a}{x} =-\frac{1}{a} ln \Big|\frac{a+r}{x} \Big|} {TEX}@@ |
| | @@||{TEX()} {s=\sqrt{x^2-a^2}} {TEX}||@@ | | @@||{TEX()} {s=\sqrt{x^2-a^2}} {TEX}||@@ |
| | @@{TEX()} {\int xs dx=\frac{1}{3} s^3} {TEX}@@ | | @@{TEX()} {\int xs dx=\frac{1}{3} s^3} {TEX}@@ |
| | @@{TEX()} {\int \frac{s \ dx}{x}=s-acos^{-1} \Big|\frac{a}{x} \Big|} {TEX}@@ | | @@{TEX()} {\int \frac{s \ dx}{x}=s-acos^{-1} \Big|\frac{a}{x} \Big|} {TEX}@@ |
| | @@{TEX()} {\int \frac{dx}{s}=ln \Big|\frac{x+s}{a} \Big|} {TEX}@@ | | @@{TEX()} {\int \frac{dx}{s}=ln \Big|\frac{x+s}{a} \Big|} {TEX}@@ |
| | --- | | --- |
| | توجه داشته باشید که : | | توجه داشته باشید که : |
| | @@{TEX()} {ln \Big|\frac{x+s}{a} \Big|=sgn(x)cosh^{-1}\Big|\frac{x}{a} \Big|=\frac{1}{2} ln \Big(\frac{x+s}{x-s} \Big) \qquad cosh^{-1}\Big| \frac{x}{a} \Big|>0} {TEX}@@ | | @@{TEX()} {ln \Big|\frac{x+s}{a} \Big|=sgn(x)cosh^{-1}\Big|\frac{x}{a} \Big|=\frac{1}{2} ln \Big(\frac{x+s}{x-s} \Big) \qquad cosh^{-1}\Big| \frac{x}{a} \Big|>0} {TEX}@@ |
| | --- | | --- |
| | @@{TEX()} {\int \frac{x \ dx}{s}=s} {TEX}@@ | | @@{TEX()} {\int \frac{x \ dx}{s}=s} {TEX}@@ |
| | @@{TEX()} {\int\frac{x \ dx}{s^3}=-\frac{1}{s}} {TEX}@@ | | @@{TEX()} {\int\frac{x \ dx}{s^3}=-\frac{1}{s}} {TEX}@@ |
| | @@{TEX()} {\int \frac{x \ dx}{s^5}=-\frac{1}{3s^3}} {TEX}@@ | | @@{TEX()} {\int \frac{x \ dx}{s^5}=-\frac{1}{3s^3}} {TEX}@@ |
| | @@{TEX()} {\int \frac{x \ dx}{s^7}=-\frac{1}{5s^5}} {TEX}@@ | | @@{TEX()} {\int \frac{x \ dx}{s^7}=-\frac{1}{5s^5}} {TEX}@@ |
| | @@{TEX()} { \int \frac{x \ dx}{s^{2n+1}}=-\frac{1}{(2n-1)s^{2n-1}}} {TEX}@@ | | @@{TEX()} { \int \frac{x \ dx}{s^{2n+1}}=-\frac{1}{(2n-1)s^{2n-1}}} {TEX}@@ |
| | @@{TEX()} { \int \frac{x^{2m}dx}{s^{2n+1}}=-\frac{x^{2m-1}}{(2n-1)s^{2n-1}}+\frac{2m-1}{2n-1} \int \frac{x^{2m-2}dx}{s^{2n-1}}} {TEX}@@ | | @@{TEX()} { \int \frac{x^{2m}dx}{s^{2n+1}}=-\frac{x^{2m-1}}{(2n-1)s^{2n-1}}+\frac{2m-1}{2n-1} \int \frac{x^{2m-2}dx}{s^{2n-1}}} {TEX}@@ |
| | @@{TEX()} {\int \frac{x^2dx}{s}=\frac{xs}{2}+\frac{a^2}{2} ln \Big|\frac{x+s}{a} \Big|} {TEX}@@ | | @@{TEX()} {\int \frac{x^2dx}{s}=\frac{xs}{2}+\frac{a^2}{2} ln \Big|\frac{x+s}{a} \Big|} {TEX}@@ |
| | @@{TEX()} {\int \frac{x^2dx}{s^3}=-\frac{x}{s}+ln \Big|\frac{x+s}{a} \Big|} {TEX}@@ | | @@{TEX()} {\int \frac{x^2dx}{s^3}=-\frac{x}{s}+ln \Big|\frac{x+s}{a} \Big|} {TEX}@@ |
| | @@{TEX()} {\int \frac{x^4 dx}{s}=\frac{x^3s}{4}+\frac{3}{8}a^2xs+\frac{3}{8}a^4 ln \Big| \frac{x+s}{a} \Big|} {TEX}@@ | | @@{TEX()} {\int \frac{x^4 dx}{s}=\frac{x^3s}{4}+\frac{3}{8}a^2xs+\frac{3}{8}a^4 ln \Big| \frac{x+s}{a} \Big|} {TEX}@@ |
| | @@{TEX()} {\int \frac{x^4dx}{s^3}=\frac{xs}{2}-\frac{a^2 x}{s} +\frac{3}{2}a^2 ln \Big|\frac{x+s}{a} \Big|} {TEX}@@ | | @@{TEX()} {\int \frac{x^4dx}{s^3}=\frac{xs}{2}-\frac{a^2 x}{s} +\frac{3}{2}a^2 ln \Big|\frac{x+s}{a} \Big|} {TEX}@@ |
| | @@{TEX()} {\int \frac{x^4 dx}{s^5}=-\frac{x}{s}-\frac{1}{3}\frac{x^3}{s^3} +ln \Big| \frac{x+s}{a} \Big|} {TEX}@@ | | @@{TEX()} {\int \frac{x^4 dx}{s^5}=-\frac{x}{s}-\frac{1}{3}\frac{x^3}{s^3} +ln \Big| \frac{x+s}{a} \Big|} {TEX}@@ |
| | @@{TEX()} {\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 (n>m \ge 0)} {TEX}@@ | | @@{TEX()} {\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 (n>m \ge 0)} {TEX}@@ |
| | @@{TEX()} {\int \frac{dx}{s^3}=-\frac{x}{a^2 s}} {TEX}@@ | | @@{TEX()} {\int \frac{dx}{s^3}=-\frac{x}{a^2 s}} {TEX}@@ |
| | @@{TEX()} {\int \frac{dx}{s^5}=\frac{1}{a^4}\Big[\frac{x}{s}-\frac{1}{3} \frac{x^3}{s^3} \Big]} {TEX}@@ | | @@{TEX()} {\int \frac{dx}{s^5}=\frac{1}{a^4}\Big[\frac{x}{s}-\frac{1}{3} \frac{x^3}{s^3} \Big]} {TEX}@@ |
| | @@{TEX()} {\int \frac{dx}{s^7}=-\frac{1}{a^6} \Big[\frac{x}{s}-\frac{2x^3}{3s^3}+\frac{x^5}{5s^5} \Big]} {TEX}@@ | | @@{TEX()} {\int \frac{dx}{s^7}=-\frac{1}{a^6} \Big[\frac{x}{s}-\frac{2x^3}{3s^3}+\frac{x^5}{5s^5} \Big]} {TEX}@@ |
| | @@{TEX()} {\int \frac{dx}{s^9}=\frac{1}{a^8}\Big[ \frac{x}{s}-\frac{3x^3}{3s^3}+\frac{3x^5}{5s^5}-\frac{x^7}{7s^7} \Big]} {TEX}@@ | | @@{TEX()} {\int \frac{dx}{s^9}=\frac{1}{a^8}\Big[ \frac{x}{s}-\frac{3x^3}{3s^3}+\frac{3x^5}{5s^5}-\frac{x^7}{7s^7} \Big]} {TEX}@@ |
| | @@{TEX()} {\int \frac{x^2 dx}{s^5}=-\frac{x^3}{3a^2s^3}} {TEX}@@ | | @@{TEX()} {\int \frac{x^2 dx}{s^5}=-\frac{x^3}{3a^2s^3}} {TEX}@@ |
| | @@{TEX()} {\int \frac{x^2 dx}{s^7}= \frac{1}{a^4} \Big[ \frac{x^3}{3s^3}-\frac{x^5}{5s^5} \Big]} {TEX}@@ | | @@{TEX()} {\int \frac{x^2 dx}{s^7}= \frac{1}{a^4} \Big[ \frac{x^3}{3s^3}-\frac{x^5}{5s^5} \Big]} {TEX}@@ |
| | @@{TEX()} {\int \frac{x^2dx}{s^9}=-\frac{1}{a^6}\Big[\frac{x^3}{3s^3}-\frac{2x^5}{5s^5}+\frac{x^7}{7s^7} \Big]} {TEX}@@ | | @@{TEX()} {\int \frac{x^2dx}{s^9}=-\frac{1}{a^6}\Big[\frac{x^3}{3s^3}-\frac{2x^5}{5s^5}+\frac{x^7}{7s^7} \Big]} {TEX}@@ |
| | @@||{TEX()} {t=\sqrt{a^2-x^2}} {TEX}||@@ | | @@||{TEX()} {t=\sqrt{a^2-x^2}} {TEX}||@@ |
| | @@{TEX()} {\int tdx=\frac{1}{2}\big(xt+a^2sin^{-1}\frac{x}{a} \big) \qquad (|x|\le |a|)} {TEX}@@ | | @@{TEX()} {\int tdx=\frac{1}{2}\big(xt+a^2sin^{-1}\frac{x}{a} \big) \qquad (|x|\le |a|)} {TEX}@@ |
| | @@{TEX()} {\int xt dx=-\frac{1}{3}t^3 \qquad (|x|\le |a|)} {TEX}@@ | | @@{TEX()} {\int xt dx=-\frac{1}{3}t^3 \qquad (|x|\le |a|)} {TEX}@@ |
| | @@{TEX()} {\int \frac{t \ dx}{x}=t-a ln \Big|\frac{a+t}{x} \Big| \qquad (|x| \le |a| )} {TEX}@@ | | @@{TEX()} {\int \frac{t \ dx}{x}=t-a ln \Big|\frac{a+t}{x} \Big| \qquad (|x| \le |a| )} {TEX}@@ |
| | @@{TEX()} {\int \frac{dx}{t}=sin^{-1} \frac{x}{a} \qquad (|x|\le |a| )} {TEX}@@ | | @@{TEX()} {\int \frac{dx}{t}=sin^{-1} \frac{x}{a} \qquad (|x|\le |a| )} {TEX}@@ |
| | @@{TEX()} {\int \frac{x^2 dx}{t}=-\frac{x}{2}t+\frac{a^2}{2}sin^{-1}\frac{x}{a} \qquad (|x|\le |a|)} {TEX}@@ | | @@{TEX()} {\int \frac{x^2 dx}{t}=-\frac{x}{2}t+\frac{a^2}{2}sin^{-1}\frac{x}{a} \qquad (|x|\le |a|)} {TEX}@@ |
| | @@{TEX()} {\int t \ dx = \frac{1}{2}\Big(xt-sgn(x)cosh^{-1} \Big|\frac{x}{a} \Big| \Big) \qquad (|x|\ge |a|)} {TEX}@@ | | @@{TEX()} {\int t \ dx = \frac{1}{2}\Big(xt-sgn(x)cosh^{-1} \Big|\frac{x}{a} \Big| \Big) \qquad (|x|\ge |a|)} {TEX}@@ |
| | @@||{TEX()} {R^{1/2}=\sqrt{ax^2+bx+c}} {TEX}||@@ | | @@||{TEX()} {R^{1/2}=\sqrt{ax^2+bx+c}} {TEX}||@@ |
| | @@{TEX()} {\int \frac{dx}{\sqrt{ax^2+bx+c}}=\frac{1}{\sqrt{a}}ln \big|2\sqrt{aR}+2ax+b \big| \qquad (a>0)} {TEX}@@ | | @@{TEX()} {\int \frac{dx}{\sqrt{ax^2+bx+c}}=\frac{1}{\sqrt{a}}ln \big|2\sqrt{aR}+2ax+b \big| \qquad (a>0)} {TEX}@@ |
| | @@{TEX()} {\int \frac{dx}{\sqrt{ax^2+bx+c}}=\frac{1}{\sqrt{a}}sinh^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}} \qquad (a>0 , 4ac-b^2>0)} {TEX}@@ | | @@{TEX()} {\int \frac{dx}{\sqrt{ax^2+bx+c}}=\frac{1}{\sqrt{a}}sinh^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}} \qquad (a>0 , 4ac-b^2>0)} {TEX}@@ |
| | @@{TEX()} {\int \frac{dx}{\sqrt{ax^2+bx+c}}=\frac{1}{\sqrt{a}}ln|2ax+b| \qquad (a>0,4ac-b^2=0)} {TEX}@@ | | @@{TEX()} {\int \frac{dx}{\sqrt{ax^2+bx+c}}=\frac{1}{\sqrt{a}}ln|2ax+b| \qquad (a>0,4ac-b^2=0)} {TEX}@@ |
| | @@{TEX()} {\int \frac{dx}{\sqrt{ax^2+bx+c}}=-\frac{1}{\sqrt{-a}}arcsin \frac{2ax+b}{\sqrt{b^2-4ac}} \qquad (a<0 ,4ac-b^2<0)} {TEX}@@ | | @@{TEX()} {\int \frac{dx}{\sqrt{ax^2+bx+c}}=-\frac{1}{\sqrt{-a}}arcsin \frac{2ax+b}{\sqrt{b^2-4ac}} \qquad (a<0 ,4ac-b^2<0)} {TEX}@@ |
| | @@{TEX()} {\int \frac{dx}{\sqrt{(ax^2+bx+c)^3}}=\frac{4ax+2b}{(4ac-b^2)\sqrt{R}}} {TEX}@@ | | @@{TEX()} {\int \frac{dx}{\sqrt{(ax^2+bx+c)^3}}=\frac{4ax+2b}{(4ac-b^2)\sqrt{R}}} {TEX}@@ |
| | @@{TEX()} {\int \frac{dx}{\sqrt{(ax^2+bx+c)^5}}=\frac{4ax+2b}{3(4ac-b^2)\sqrt{R}} \Big( \frac{1}{R}+\frac{8a}{4ac-b^2} \Big)} {TEX}@@ | | @@{TEX()} {\int \frac{dx}{\sqrt{(ax^2+bx+c)^5}}=\frac{4ax+2b}{3(4ac-b^2)\sqrt{R}} \Big( \frac{1}{R}+\frac{8a}{4ac-b^2} \Big)} {TEX}@@ |
| | @@{TEX()} {\int \frac{dx}{\sqrt{(ax^2+bx+c)^{2n+1}}}=\frac{4ax+2b}{(2n-1)(4ac-b^2)R^{(2n-1)/2}}+\frac{8a(n-1)}{(2n-1)(4ac-b^2)} \int \frac{dx}{R^{(2n-1)/2}}} {TEX}@@ | | @@{TEX()} {\int \frac{dx}{\sqrt{(ax^2+bx+c)^{2n+1}}}=\frac{4ax+2b}{(2n-1)(4ac-b^2)R^{(2n-1)/2}}+\frac{8a(n-1)}{(2n-1)(4ac-b^2)} \int \frac{dx}{R^{(2n-1)/2}}} {TEX}@@ |
| | @@{TEX()} {\int \frac{x \ dx}{\sqrt{ax^2+bx+c}}=\frac{\sqrt{R}}{a}-\frac{b}{2a} \int {\frac{dx}{\sqrt{R}}} {TEX}@@ | | @@{TEX()} {\int \frac{x \ dx}{\sqrt{ax^2+bx+c}}=\frac{\sqrt{R}}{a}-\frac{b}{2a} \int {\frac{dx}{\sqrt{R}}} {TEX}@@ |
| | @@{TEX()} {\int \frac{x \ dx}{\sqrt{(ax^2+bx+c)^3}}=-\frac{2bx+4c}{(4ac-b^2)\sqrt{R}}} {TEX}@@ | | @@{TEX()} {\int \frac{x \ dx}{\sqrt{(ax^2+bx+c)^3}}=-\frac{2bx+4c}{(4ac-b^2)\sqrt{R}}} {TEX}@@ |
| | @@{TEX()} {\int \frac{x \ dx}{\sqrt{(ax^2+bx+c)^{2n+1}}}=-\frac{1}{(2n-1)aR^{(2n-1)/2}}-\frac{b}{2a} \int \frac{ dx}{R^{(2n+1)/2}} {TEX}@@ | | @@{TEX()} {\int \frac{x \ dx}{\sqrt{(ax^2+bx+c)^{2n+1}}}=-\frac{1}{(2n-1)aR^{(2n-1)/2}}-\frac{b}{2a} \int \frac{ dx}{R^{(2n+1)/2}} {TEX}@@ |
| | @@{TEX()} {\int \frac{dx}{x \sqrt{ax^2+bx+c}}=-\frac{1}{\sqrt{c}}ln \Big( \frac{2\sqrt{cR}+bx+2c}{x} \Big)=-\frac{1}{\sqrt{c}}sinh^{-1} \Big( \frac{bx+2c}{|x| \sqrt{4ac-b^2}} \Big)} {TEX}@@ | | @@{TEX()} {\int \frac{dx}{x \sqrt{ax^2+bx+c}}=-\frac{1}{\sqrt{c}}ln \Big( \frac{2\sqrt{cR}+bx+2c}{x} \Big)=-\frac{1}{\sqrt{c}}sinh^{-1} \Big( \frac{bx+2c}{|x| \sqrt{4ac-b^2}} \Big)} {TEX}@@ |
| | @@||{TEX()} {R^{1/2}=\sqrt{ax+b}} {TEX}||@@ | | @@||{TEX()} {R^{1/2}=\sqrt{ax+b}} {TEX}||@@ |
| | @@{TEX()} {\int \frac{dx}{x \sqrt{ax+b}}=\frac{-2}{\sqrt{b}}tanh^{-1} \sqrt {\frac{ax+b}{b}}} {TEX}@@ | | @@{TEX()} {\int \frac{dx}{x \sqrt{ax+b}}=\frac{-2}{\sqrt{b}}tanh^{-1} \sqrt {\frac{ax+b}{b}}} {TEX}@@ |
| | @@{TEX()} {\int \frac{\sqrt{ax+b}}{x} dx=2Big(\sqrt{ax+b}-\sqer{b} tanh^{-1} \sqrt{\frac{ax+b}{b}} \Big)} {TEX}@@ | | @@{TEX()} {\int \frac{\sqrt{ax+b}}{x} dx=2Big(\sqrt{ax+b}-\sqer{b} tanh^{-1} \sqrt{\frac{ax+b}{b}} \Big)} {TEX}@@ |
| | @@{TEX()} {\int \frac{x^n}{\sqrt{ax+b}}dx=\frac{2}{a(2n+1)} \Big(x^n \sqrt{ax+b}-bn \int \frac{x^{n-1}}{\sqrt{ax+b}} \Big)} {TEX}@@ | | @@{TEX()} {\int \frac{x^n}{\sqrt{ax+b}}dx=\frac{2}{a(2n+1)} \Big(x^n \sqrt{ax+b}-bn \int \frac{x^{n-1}}{\sqrt{ax+b}} \Big)} {TEX}@@ |
| | @@{TEX()} {\int x^n \sqrt{ax+b}dx =\frac{2}{2n+1} \big(x^{n+1} \sqrt{ax+b} +bx^n \sqrt{ax+b}-nb \int x^{n-1} \sqrt{ax+b}dx \big)} {TEX}@@ | | @@{TEX()} {\int x^n \sqrt{ax+b}dx =\frac{2}{2n+1} \big(x^{n+1} \sqrt{ax+b} +bx^n \sqrt{ax+b}-nb \int x^{n-1} \sqrt{ax+b}dx \big)} {TEX}@@ |
| | --- | | --- |
| | !همچنین ببینید: | | !همچنین ببینید: |
| | *((جدول انتگرال توابع گویا)) | | *((جدول انتگرال توابع گویا)) |
| | *((جدول انتگرال توابع لگاریتمی)) | | *((جدول انتگرال توابع لگاریتمی)) |
| | *((جدول انتگرال توابع نمایی)) | | *((جدول انتگرال توابع نمایی)) |
| | *((جدول انتگرال توابع مثلثاتی)) | | *((جدول انتگرال توابع مثلثاتی)) |
| | + | *((جدول انتگرال معکوس توابع مثلثاتی)) |
| | *((جدول انتگرال توابع هیپربولیک)) | | *((جدول انتگرال توابع هیپربولیک)) |
| | #@^ | | #@^ |
