以下是部份無理函式的積分表。書寫時省略了不定積分結果中都含有的任意常數 Cn 包括 r a2 x2 的積分 int r dx frac 12left x r a2lnleft frac xrarightright int r3 dx frac 14 xr3frac 18 3a2xrfrac 38 a4lnleft frac xraright int r5 dx frac 16 xr5frac 524 a2xr3frac 516 a4xrfrac 516 a6lnleft frac xraright int x rdxfrac r33 int x r3dxfrac r55 int x r2n1dxfrac r2n32n3 int x2 rdx frac xr34frac a2xr8frac a48lnleft frac xraright int x2 r3dx frac xr56frac a2xr324frac a4xr16frac a616lnleft frac xraright int x3 r dx frac r55 frac a2 r33 int x3 r3 dx frac r77frac a2r55 int x3 r2n1 dx frac r2n52n5 frac a2 r2n32n3 int x4 rdx frac x3r36frac a2xr38frac a4xr16frac a616lnleft frac xraright int x4 r3dx frac x3r58frac a2xr516frac a4xr364frac 3a6xr128frac 3a8128lnleft frac xraright int x5 r dx frac r77 frac 2 a2 r55 frac a4 r33 int x5 r3 dx frac r99 frac 2 a2 r77 frac a4 r55 int x5 r2n1 dx frac r2n72n7 frac 2a2r2n52n5frac a4 r2n32n3 intfrac rdxx ralnleftfrac arxright r a sinh1frac ax intfrac r3dxx frac r33a2ra3lnleftfrac arxright intfrac r5dxx frac r55frac a2r33a4ra5lnleftfrac arxright intfrac r7dxx frac r77frac a2r55frac a4r33a6ra7lnleftfrac arxright intfrac dxr sinh1frac xa lnleftxrright intfrac xdxr r intfrac x2dxr frac x2 rfrac a22sinh1frac xa frac x2 rfrac a22lnleftxrright intfrac dxxr frac 1asinh1frac ax frac 1alnleftfrac arxright 包括 s x2 a2 的積分 假設 x2a2 x2a2 請看下一節 int xsdx frac 13 s3 intfrac sdxx s acos1leftfrac axright intfrac dxs intfrac dxsqrt x2a2 lnleftxsright 注意在 lnleftfrac xsaright mathrm sgnxcosh1leftfrac xaright frac 12lnleft frac xsxsright 裡 cosh1leftfrac xaright 取正值 intfrac xdxs s intfrac xdxs3 frac 1s intfrac xdxs5 frac 13s3 intfrac xdxs7 frac 15s5 intfrac xdxs2n1 frac 12n1 s2n1 intfrac x2mdxs2n1 frac 12n1frac x2m1s2n1frac 2m12n1intfrac x2m2dxs2n1 intfrac x2dxs frac xs2frac a22lnleftfrac xsaright intfrac x2dxs3 frac xslnleftfrac xsaright intfrac x4dxs frac x3s4frac 38 a2xsfrac 38 a4lnleftfrac xsaright intfrac x4dxs3 frac xs2frac a2xsfrac 32 a2lnleftfrac xsaright intfrac x4dxs5 frac xsfrac 13frac x3s3lnleftfrac xsaright intfrac x2mdxs2n1 1nmfrac 1a2 nmsum_i0nm1frac 12 mi1nm1 choose ifrac x2 mi1s2 mi1qquadmbox nmge0mbox intfrac dxs3frac 1a2frac xs intfrac dxs5frac 1a4left frac xsfrac 13frac x3s3right intfrac dxs7 frac 1a6left frac xsfrac 23frac x3s3frac 15frac x5s5right intfrac dxs9 frac 1a8left frac xsfrac 33frac x3s3frac 35frac x5s5frac 17frac x7s7right intfrac x2dxs5frac 1a2frac x33s3 intfrac x2dxs7 frac 1a4left frac 13frac x3s3frac 15frac x5s5right intfrac x2dxs9 frac 1a6left frac 13frac x3s3frac 25frac x5s5frac 17frac x7s7right 包括 t a2 x2 的積分 int t dx frac 12left xta2sin1frac xaright qquadmbox xleqambox int xtdx frac 13 t3 qquadmbox xleqambox intfrac tdxx talnleftfrac atxright qquadmbox xleqambox intfrac dxt sin1frac xa qquadmbox xleqambox intfrac x2dxt frac x2 tfrac a22sin1frac xa qquadmbox xleqambox int tdx frac 12left xtsgn xcosh1leftfrac xarightright qquadmbox xgeambox int xnsqrt 1x2text d xfrac left xn1xn1 right sqrt 1x2n1frac n1n1int xn2sqrt 1x2text d x 包括R ax2 bx c 的積分 intfrac dxsqrt ax2bxc frac 1sqrt alnleft2sqrt a R2axbright qquadqquad mbox a0mbox intfrac dxsqrt ax2bxc frac 1sqrt asinh1frac 2axbsqrt 4acb2 qquadqquad mbox a0mbox 4acb20mbox intfrac dxsqrt ax2bxc frac 1sqrt aln2axb quad mbox a0mbox 4acb20mbox intfrac dxsqrt ax2bxc frac 1sqrt aarcsinfrac 2axbsqrt b24ac qquadqquad mbox a0mbox 4acb20mbox intfrac dxsqrt ax2bxc3 frac 4ax2b4acb2sqrt R intfrac dxsqrt ax2bxc5 frac 4ax2b3 4acb2sqrt Rleft frac 1Rfrac 8a4acb2right intfrac dxsqrt ax2bxc2n1 frac 4ax2b2n14acb2 R2n12frac 8a n12n14acb2 intfrac dxR2n12 intfrac xdxsqrt ax2bxc frac sqrt Rafrac b2aintfrac dxsqrt R intfrac xdxsqrt ax2bxc3 frac 2bx4c4acb2sqrt R intfrac xdxsqrt ax2bxc2n1 frac 12n1 aR2n12frac b2aintfrac dxR2n12 intfrac dxxsqrt ax2bxcfrac 1sqrt clnleft frac 2sqrt c Rbx2cxright intfrac dxxsqrt ax2bxcfrac 1sqrt csinh1left frac bx2cxsqrt 4acb2right 包括 R ax b 的積分 int frac dxxsqrt ax bfrac 2sqrt btanh1sqrt frac ax bb intfrac sqrt ax bxdx2left sqrt ax b sqrt btanh1sqrt frac ax bbright intfrac xnsqrt ax bdxfrac 2a 2n1 left xnsqrt ax b bnintfrac xn1sqrt ax bright int xn sqrt ax bdx frac 22n 1left xn1 sqrt ax b bxn sqrt ax b nbint xn1sqrt ax bdx right Category 積分表