無理代数関数の積分の一覧JJapedia 編集部|更新日: 2026年7月26日以下は、無理関数の積分（原始関数） の一覧です。積分関数の完全な一覧については、「積分の一覧」を参照してください。この記事全体を通して、積分変数およびすべてのパラメータは実数であると仮定し、簡潔にするために積分定数は省略しています。r = √ a 2 + x 2を含む積分∫rdx=12（xr+12ln（x+r））{\displaystyle \int r\,dx={\frac {1}{2}}\left(xr+a^{2}\,\ln \left(x+r\right)\right)}∫r3dx=14xr3+3812xr+3814ln（x+r）{\displaystyle \int r^{3}\,dx={\frac {1}{4}}xr^{3}+{\frac {3}{8}}a^{2}xr+{\frac {3}{8}}a^{4}\ln \left(x+r\right)}∫r5dx=16xr5+52412xr3+51614xr+51616ln（x+r）{\displaystyle \int r^{5}\,dx={\frac {1}{6}}xr^{5}+{\frac {5}{24}}a^{2}xr^{3}+{\frac {5}{16}}a^{4}xr+{\frac {5}{16}}a^{6}\ln \left(x+r\right)}∫xrdx=r33{\displaystyle \int xr\,dx={\frac {r^{3}}{3}}}∫xr3dx=r55{\displaystyle \int xr^{3}\,dx={\frac {r^{5}}{5}}}∫xr2n+1dx=r2n+32n+3{\displaystyle \int xr^{2n+1}\,dx={\frac {r^{2n+3}}{2n+3}}}∫x2rdx=xr34−12xr8−148ln（x+r）{\displaystyle \int x^{2}r\,dx={\frac {xr^{3}}{4}}-{\frac {a^{2}xr}{8}}-{\frac {a^{4}}{8}}\ln \left(x+r\right)}∫x2r3dx=xr56−12xr324−14xr16−1616ln（x+r）{\displaystyle \int x^{2}r^{3}\,dx={\frac {xr^{5}}{6}}-{\frac {a^{2}xr^{3}}{24}}-{\frac {a^{4}xr}{16}}-{\frac {a^{6}}{16}}\ln \left(x+r\right)}∫x3rdx=r55−12r33{\displaystyle \int x^{3}r\,dx={\frac {r^{5}}{5}}-{\frac {a^{2}r^{3}}{3}}}∫x3r3dx=r77−12r55{\displaystyle \int x^{3}r^{3}\,dx={\frac {r^{7}}{7}}-{\frac {a^{2}r^{5}}{5}}}∫x3r2n+1dx=r2n+52n+5−12r2n+32n+3{\displaystyle \int x^{3}r^{2n+1}\,dx={\frac {r^{2n+5}}{2n+5}}-{\frac {a^{2}r^{2n+3}}{2n+3}}}∫x4rdx=x3r36−12xr38+14xr16+1616ln（x+r）{\displaystyle \int x^{4}r\,dx={\frac {x^{3}r^{3}}{6}}-{\frac {a^{2}xr^{3}}{8}}+{\frac {a^{4}xr}{16}}+{\frac {a^{6}}{16}}\ln \left(x+r\right)}∫x4r3dx=x3r58−12xr516+14xr364+316xr128+318128ln（x+r）{\displaystyle \int x^{4}r^{3}\,dx={\frac {x^{3}r^{5}}{8}}-{\frac {a^{2}xr^{5}}{16}}+{\frac {a^{4}xr^{3}}{64}}+{\frac {3a^{6}xr}{128}}+{\frac {3a^{8}}{128}}\ln \left(x+r\right)}∫x5rdx=r77−212r55+14r33{\displaystyle \int x^{5}r\,dx={\frac {r^{7}}{7}}-{\frac {2a^{2}r^{5}}{5}}+{\frac {a^{4}r^{3}}{3}}}∫x5r3dx=r99−212r77+14r55{\displaystyle \int x^{5}r^{3}\,dx={\frac {r^{9}}{9}}-{\frac {2a^{2}r^{7}}{7}}+{\frac {a^{4}r^{5}}{5}}}∫x5r2n+1dx=r2n+72n+7−212r2n+52n+5+14r2n+32n+3{\displaystyle \int x^{5}r^{2n+1}\,dx={\frac {r^{2n+7}}{2n+7}}-{\frac {2a^{2}r^{2n+5}}{2n+5}}+{\frac {a^{4}r^{2n+3}}{2n+3}}}∫rdxx=r−1ln|1+rx|=r−1アルシン1x{\displaystyle \int {\frac {r\,dx}{x}}=ra\ln \left|{\frac {a+r}{x}}\right|=ra\,\operatorname {arsinh} {\frac {a}{x}}}∫r3dxx=r33+12r−13ln|1+rx|{\displaystyle \int {\frac {r^{3}\,dx}{x}}={\frac {r^{3}}{3}}+a^{2}r-a^{3}\ln \left|{\frac {a+r}{x}}\right|}∫r5dxx=r55+12r33+14r−15ln|1+rx|{\displaystyle \int {\frac {r^{5}\,dx}{x}}={\frac {r^{5}}{5}}+{\frac {a^{2}r^{3}}{3}}+a^{4}r-a^{5}\ln \left|{\frac {a+r}{x}}\right|}∫r7dxx=r77+12r55+14r33+16r−17ln|1+rx|{\displaystyle \int {\frac {r^{7}\,dx}{x}}={\frac {r^{7}}{7}}+{\frac {a^{2}r^{5}}{5}}+{\frac {a^{4}r^{3}}{3}}+a^{6}r-a^{7}\ln \left|{\frac {a+r}{x}}\right|}∫dxr=アルシンx1=ln（x+r1）{\displaystyle \int {\frac {dx}{r}}=\operatorname {arsinh} {\frac {x}{a}}=\ln \left({\frac {x+r}{a}}\right)}∫dxr3=x12r{\displaystyle \int {\frac {dx}{r^{3}}}={\frac {x}{a^{2}r}}}∫xdxr=r{\displaystyle \int {\frac {x\,dx}{r}}=r}∫xdxr3=−1r{\displaystyle \int {\frac {x\,dx}{r^{3}}}=-{\frac {1}{r}}}∫x2dxr=x2r−122アルシンx1=x2r−122ln（x+r1）{\displaystyle \int {\frac {x^{2}\,dx}{r}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\,\operatorname {arsinh} {\frac {x}{a}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\ln \left({\frac {x+r}{a}}\right)}∫dxxr=−11アルシン1x=−11ln|1+rx|{\displaystyle \int {\frac {dx}{xr}}=-{\frac {1}{a}}\,\operatorname {arsinh} {\frac {a}{x}}=-{\frac {1}{a}}\ln \left|{\frac {a+r}{x}}\right|}s = √ x 2 − a 2を含む積分x 2 > a 2と仮定します（x 2 < a 2の場合は次のセクションを参照してください）。∫sdx=12（xs−12ln|x+s|）{\displaystyle \int s\,dx={\frac {1}{2}}\left(xs-a^{2}\ln \left|x+s\right|\right)}∫xsdx=13s3{\displaystyle \int xs\,dx={\frac {1}{3}}s^{3}}∫sdxx=s−|1|アルコス|1x|{\displaystyle \int {\frac {s\,dx}{x}}=s-|a|\arccos \left|{\frac {a}{x}}\right|}∫dxs=ln|x+s1|=サイン（x）アルコッシュ|x1|=12ln（x+sx−s）、{\displaystyle \int {\frac {dx}{s}}=\ln \left|{\frac {x+s}{a}}\right|=\operatorname {sgn} (x)\,\operatorname {arcosh} \left|{\frac {x}{a}}\right|={\frac {1}{2}}\ln \left({\frac {x+s}{x-s}}\right)\,,}ここで、正の値はアルコッシュ|x1|{\displaystyle \operatorname {arcosh} \left|{\frac {x}{a}}\right|}服用する必要があります。∫dxxs=11アークセグ|x1|{\displaystyle \int {\frac {dx}{xs}}={\frac {1}{a}}\operatorname {arcsec} \left|{\frac {x}{a}}\right|}∫xdxs=s{\displaystyle \int {\frac {x\,dx}{s}}=s}∫xdxs3=−1s{\displaystyle \int {\frac {x\,dx}{s^{3}}}=-{\frac {1}{s}}}∫xdxs5=−13s3{\displaystyle \int {\frac {x\,dx}{s^{5}}}=-{\frac {1}{3s^{3}}}}∫xdxs7=−15s5{\displaystyle \int {\frac {x\,dx}{s^{7}}}=-{\frac {1}{5s^{5}}}}∫xdxs2n+1=−1（2n−1）s2n−1{\displaystyle \int {\frac {x\,dx}{s^{2n+1}}}=-{\frac {1}{(2n-1)s^{2n-1}}}}∫x2mdxs2n+1=−12n−1x2m−1s2n−1+2m−12n−1∫x2m−2dxs2n−1{\displaystyle \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}}}}∫x2dxs=xs2+122ln|x+s1|{\displaystyle \int {\frac {x^{2}\,dx}{s}}={\frac {xs}{2}}+{\frac {a^{2}}{2}}\ln \left|{\frac {x+s}{a}}\right|}∫x2dxs3=−xs+ln|x+s1|{\displaystyle \int {\frac {x^{2}\,dx}{s^{3}}}=-{\frac {x}{s}}+\ln \left|{\frac {x+s}{a}}\right|}∫x4dxs=x3s4+3812xs+3814ln|x+s1|{\displaystyle \int {\frac {x^{4}\,dx}{s}}={\frac {x^{3}s}{4}}+{\frac {3}{8}}a^{2}xs+{\frac {3}{8}}a^{4}\ln \left|{\frac {x+s}{a}}\right|}∫x4dxs3=xs2−12xs+3212ln|x+s1|{\displaystyle \int {\frac {x^{4}\,dx}{s^{3}}}={\frac {xs}{2}}-{\frac {a^{2}x}{s}}+{\frac {3}{2}}a^{2}\ln \left|{\frac {x+s}{a}}\right|}∫x4dxs5=−xs−13x3s3+ln|x+s1|{\displaystyle \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|}∫x2mdxs2n+1=（−1）n−m112（n−m）∑私=0n−m−112（m+私）+1（n−m−1私）x2（m+私）+1s2（m+私）+1（n>m≥0）{\displaystyle \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\geq 0{\mbox{)}}}∫dxs3=−112xs{\displaystyle \int {\frac {dx}{s^{3}}}=-{\frac {1}{a^{2}}}{\frac {x}{s}}}∫dxs5=114[xs−13x3s3]{\displaystyle \int {\frac {dx}{s^{5}}}={\frac {1}{a^{4}}}\left[{\frac {x}{s}}-{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}\right]}∫dxs7=−116[xs−23x3s3+15x5s5]{\displaystyle \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]}∫dxs9=118[xs−33x3s3+35x5s5−17x7s7]{\displaystyle \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]}∫x2dxs5=−112x33s3{\displaystyle \int {\frac {x^{2}\,dx}{s^{5}}}=-{\frac {1}{a^{2}}}{\frac {x^{3}}{3s^{3}}}}∫x2dxs7=114[13x3s3−15x5s5]{\displaystyle \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]}∫x2dxs9=−116[13x3s3−25x5s5+17x7s7]{\displaystyle \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]}u = √ a 2 − x 2を含む積分∫udx=12（xu+12arcsinx1）（|x|≤|1|）{\displaystyle \int u\,dx={\frac {1}{2}}\left(xu+a^{2}\arcsin {\frac {x}{a}}\right)\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}∫xudx=−13u3（|x|≤|1|）{\displaystyle \int xu\,dx=-{\frac {1}{3}}u^{3}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}∫x2udx=−x4u3+128（xu+12arcsinx1）（|x|≤|1|）{\displaystyle \int x^{2}u\,dx=-{\frac {x}{4}}u^{3}+{\frac {a^{2}}{8}}(xu+a^{2}\arcsin {\frac {x}{a}})\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}∫udxx=u−1ln|1+ux|（|x|≤|1|）{\displaystyle \int {\frac {u\,dx}{x}}=u-a\ln \left|{\frac {a+u}{x}}\right|\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}∫dxu=arcsinx1（|x|≤|1|）{\displaystyle \int {\frac {dx}{u}}=\arcsin {\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}∫x2dxu=12（−xu+12arcsinx1）（|x|≤|1|）{\displaystyle \int {\frac {x^{2}\,dx}{u}}={\frac {1}{2}}\left(-xu+a^{2}\arcsin {\frac {x}{a}}\right)\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}∫udx=12（xu−サインxアルコッシュ|x1|）（のために |x|≥|1|）{\displaystyle \int u\,dx={\frac {1}{2}}\left(xu-\operatorname {sgn} x\,\operatorname {arcosh} \left|{\frac {x}{a}}\right|\right)\qquad {\mbox{(for }}|x|\geq |a|{\mbox{)}}}∫xudx=−u（|x|≤|1|）{\displaystyle \int {\frac {x}{u}}\,dx=-u\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}R = √ ax 2 + bx + cを含む積分( ax 2 + bx + c ) は、あるpとqに対して次の式 ( px + q ) 2に還元できないと仮定します。∫dxR=11ln|21R+21x+b|（のために 1>0）{\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\ln \left|2{\sqrt {a}}R+2ax+b\right|\qquad {\mbox{(for }}a>0{\mbox{)}}}∫dxR=11アルシン21x+b41c−b2（のために 1>0、 41c−b2>0）{\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\,\operatorname {arsinh} {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{(for }}a>0{\mbox{, }}4ac-b^{2}>0{\mbox{)}}}∫dxR=11ln|21x+b|（のために 1>0、 41c−b2=0）{\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\ln |2ax+b|\quad {\mbox{(for }}a>0{\mbox{, }}4ac-b^{2}=0{\mbox{)}}}∫dxR=−1−1arcsin21x+bb2−41c（のために 1<0、 41c−b2<0、 |21x+b|<b2−41c）{\displaystyle \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{)}}}∫dxR3=41x+2b（41c−b2）R{\displaystyle \int {\frac {dx}{R^{3}}}={\frac {4ax+2b}{(4ac-b^{2})R}}}∫dxR5=41x+2b3（41c−b2）R（1R2+8141c−b2）{\displaystyle \int {\frac {dx}{R^{5}}}={\frac {4ax+2b}{3(4ac-b^{2})R}}\left({\frac {1}{R^{2}}}+{\frac {8a}{4ac-b^{2}}}\right)}∫dxR2n+1=2（2n−1）（41c−b2）（21x+bR2n−1+41（n−1）∫dxR2n−1）{\displaystyle \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)}∫xRdx=R1−b21∫dxR{\displaystyle \int {\frac {x}{R}}\,dx={\frac {R}{a}}-{\frac {b}{2a}}\int {\frac {dx}{R}}}∫xR3dx=−2bx+4c（41c−b2）R{\displaystyle \int {\frac {x}{R^{3}}}\,dx=-{\frac {2bx+4c}{(4ac-b^{2})R}}}∫xR2n+1dx=−1（2n−1）1R2n−1−b21∫dxR2n+1{\displaystyle \int {\frac {x}{R^{2n+1}}}\,dx=-{\frac {1}{(2n-1)aR^{2n-1}}}-{\frac {b}{2a}}\int {\frac {dx}{R^{2n+1}}}}∫dxxR=−1cln|2cR+bx+2cx|、 c>0{\displaystyle \int {\frac {dx}{xR}}=-{\frac {1}{\sqrt {c}}}\ln \left|{\frac {2{\sqrt {c}}R+bx+2c}{x}}\right|,~c>0}∫dxxR=−1cアルシン（bx+2c|x|41c−b2）、 c<0{\displaystyle \int {\frac {dx}{xR}}=-{\frac {1}{\sqrt {c}}}\operatorname {arsinh} \left({\frac {bx+2c}{|x|{\sqrt {4ac-b^{2}}}}}\right),~c<0}∫dxxR=1−carcsin（bx+2c|x|b2−41c）、 c<0、b2−41c>0{\displaystyle \int {\frac {dx}{xR}}={\frac {1}{\sqrt {-c}}}\operatorname {arcsin} \left({\frac {bx+2c}{|x|{\sqrt {b^{2}-4ac}}}}\right),~c<0,b^{2}-4ac>0}∫dxxR=−2bx（1x2+bx）、 c=0{\displaystyle \int {\frac {dx}{xR}}=-{\frac {2}{bx}}\left({\sqrt {ax^{2}+bx}}\right),~c=0}∫x2Rdx=21x−3b412R+3b2−41c812∫dxR{\displaystyle \int {\frac {x^{2}}{R}}\,dx={\frac {2ax-3b}{4a^{2}}}R+{\frac {3b^{2}-4ac}{8a^{2}}}\int {\frac {dx}{R}}}∫dxx2R=−Rcx−b2c∫dxxR{\displaystyle \int {\frac {dx}{x^{2}R}}=-{\frac {R}{cx}}-{\frac {b}{2c}}\int {\frac {dx}{xR}}}∫Rdx=21x+b41R+41c−b281∫dxR{\displaystyle \int R\,dx={\frac {2ax+b}{4a}}R+{\frac {4ac-b^{2}}{8a}}\int {\frac {dx}{R}}}∫xRdx=R331−b（21x+b）812R−b（41c−b2）1612∫dxR{\displaystyle \int xR\,dx={\frac {R^{3}}{3a}}-{\frac {b(2ax+b)}{8a^{2}}}R-{\frac {b(4ac-b^{2})}{16a^{2}}}\int {\frac {dx}{R}}}∫x2Rdx=61x−5b2412R3+5b2−41c1612∫Rdx{\displaystyle \int x^{2}R\,dx={\frac {6ax-5b}{24a^{2}}}R^{3}+{\frac {5b^{2}-4ac}{16a^{2}}}\int R\,dx}∫Rxdx=R+b2∫dxR+c∫dxxR{\displaystyle \int {\frac {R}{x}}\,dx=R+{\frac {b}{2}}\int {\frac {dx}{R}}+c\int {\frac {dx}{xR}}}∫Rx2dx=−Rx+1∫dxR+b2∫dxxR{\displaystyle \int {\frac {R}{x^{2}}}\,dx=-{\frac {R}{x}}+a\int {\frac {dx}{R}}+{\frac {b}{2}}\int {\frac {dx}{xR}}}∫x2dxR3=（2b2−41c）x+2bc1（41c−b2）R+11∫dxR{\displaystyle \int {\frac {x^{2}\,dx}{R^{3}}}={\frac {(2b^{2}-4ac)x+2bc}{a(4ac-b^{2})R}}+{\frac {1}{a}}\int {\frac {dx}{R}}}S = √ ax + bを含む積分∫Sdx=2S331{\displaystyle \int S\,dx={\frac {2S^{3}}{3a}}}∫dxS=2S1{\displaystyle \int {\frac {dx}{S}}={\frac {2S}{a}}}∫dxxS={−2bアルコット（Sb）（のために b>0、1x>0）−2bアルタン（Sb）（のために b>0、1x<0）2−bアークタン（S−b）（のために b<0）{\displaystyle \int {\frac {dx}{xS}}={\begin{cases}-{\dfrac {2}{\sqrt {b}}}\operatorname {arcoth} \left({\dfrac {S}{\sqrt {b}}}\right)&{\mbox{(for }}b>0,\quad ax>0{\mbox{)}}\\-{\dfrac {2}{\sqrt {b}}}\operatorname {artanh} \left({\dfrac {S}{\sqrt {b}}}\right)&{\mbox{(for }}b>0,\quad ax<0{\mbox{)}}\\{\dfrac {2}{\sqrt {-b}}}\arctan \left({\dfrac {S}{\sqrt {-b}}}\right)&{\mbox{(for }}b<0{\mbox{)}}\\\end{cases}}}∫Sxdx={2（S−bアルコット（Sb））（のために b>0、1x>0）2（S−bアルタン（Sb））（のために b>0、1x<0）2（S−−bアークタン（S−b））（のために b<0）{\displaystyle \int {\frac {S}{x}}\,dx={\begin{cases}2\left(S-{\sqrt {b}}\,\operatorname {arcoth} \left({\dfrac {S}{\sqrt {b}}}\right)\right)&{\mbox{(for }}b>0,\quad ax>0{\mbox{)}}\\2\left(S-{\sqrt {b}}\,\operatorname {artanh} \left({\dfrac {S}{\sqrt {b}}}\right)\right)&{\mbox{(for }}b>0,\quad ax<0{\mbox{)}}\\2\left(S-{\sqrt {-b}}\arctan \left({\dfrac {S}{\sqrt {-b}}}\right)\right)&{\mbox{(for }}b<0{\mbox{)}}\\\end{cases}}}∫xnSdx=21（2n+1）（xnS−bn∫xn−1Sdx）{\displaystyle \int {\frac {x^{n}}{S}}\,dx={\frac {2}{a(2n+1)}}\left(x^{n}S-bn\int {\frac {x^{n-1}}{S}}\,dx\right)}∫xnSdx=21（2n+3）（xnS3−nb∫xn−1Sdx）{\displaystyle \int x^{n}S\,dx={\frac {2}{a(2n+3)}}\left(x^{n}S^{3}-nb\int x^{n-1}S\,dx\right)}∫1xnSdx=−1b（n−1）（Sxn−1+（n−32）1∫dxxn−1S）{\displaystyle \int {\frac {1}{x^{n}S}}\,dx=-{\frac {1}{b(n-1)}}\left({\frac {S}{x^{n-1}}}+\left(n-{\frac {3}{2}}\right)a\int {\frac {dx}{x^{n-1}S}}\right)}参考文献アブラモウィッツ、ミルトン；ステガン、アイリーン A. 編 (1972)。「第 3 章」。数式、グラフ、および数表を含む数学関数ハンドブック。ニューヨーク：ドーバー出版。Gradshteyn, イズライル・ソロモノヴィッチ;ヨシフ・モシェヴィッチ・リジク;ジェロニムス、ユーリ・ヴェニアミノヴィッチ;ツェイトリン、ミハイル・ユリエヴィッチ;ジェフリー、アラン (2015) [2014 年 10 月]。ツウィリンガー、ダニエル。モル、ヴィクトル・ユーゴー（編）。インテグラル、シリーズ、および製品の表。 Scripta Technica, Inc. による翻訳 (第 8 版)。Academic Press, Inc. ISBN 978-0-12-384933-5。LCCN 2014010276。 （過去の版もいくつかあります。）ピアース、ベンジャミン・オズグッド (1929) [1899]。「第3章」。積分簡約表（第3版改訂 版）。ボストン：ギン・アンド・カンパニー。16-30頁。 カテゴリー：積分の一覧非表示のカテゴリ:短い説明付きの記事短い説明はWikidataとは異なります関連するトピック関連以下は、無理関数の関連積分関連原始関連「積分の一覧」を参照してください。この記事全体を通して、積分変数およびすべてのパラメータは
ここで、正の値は
服用する必要があります。
![{\displaystyle \int {\frac {dx}{s^{5}}}={\frac {1}{a^{4}}}\left[{\frac {x}{s}}-{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/054a5959ce5e03cf279c1b29dff2ba014ac6dcde)
![{\displaystyle \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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86843311de7fc72bc01f87742445f7c4b88899e9)
![{\displaystyle \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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca32b3a8d7f9040840f5d1de3467129edff0d80b)
![{\displaystyle \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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a98057cf3f3d6b7025114445c972bb6b7b7af9d)
![{\displaystyle \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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cce4b87e7a47ce42042803038139f830afd5d37)
