İrrasyonel fonksiyonların integralleri

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

r=x2+a2 içeren integraller

rdx=12(xr+a2ln(x+r))
r3dx=14xr3+183a2xr+38a4ln(x+r)
r5dx=16xr5+524a2xr3+516a4xr+516a6ln(x+r)
xrdx=r33
xr3dx=r55
xr2n+1dx=r2n+32n+3
x2rdx=xr34a2xr8a48ln(x+r)
x2r3dx=xr56a2xr324a4xr16a616ln(x+r)
x3rdx=r55a2r33
x3r3dx=r77a2r55
x3r2n+1dx=r2n+52n+5a3r2n+32n+3
x4rdx=x3r36a2xr38+a4xr16+a616ln(x+r)
x4r3dx=x3r58a2xr516+a4xr364+3a6xr128+3a8128ln(x+r)
x5rdx=r772a2r55+a4r33
x5r3dx=r992a2r77+a4r55
x5r2n+1dx=r2n+72n+72a2r2n+52n+5+a4r2n+32n+3
rdxx=raln|a+rx|=rasinh1ax
r3dxx=r33+a2ra3ln|a+rx|
r5dxx=r55+a2r33+a4ra5ln|a+rx|
r7dxx=r77+a2r55+a4r33+a6ra7ln|a+rx|
dxr=sinh1xa=ln|x+r|
dxr3=xa2r
xdxr=r
xdxr3=1r
x2dxr=x2ra22sinh1xa=x2ra22ln|x+r|
dxxr=1asinh1ax=1aln|a+rx|

s=x2a2 içeren integraller

(x2>a2) olduğunu varsayın. (x2<a2) için bir sonraki bölüme bakınız.

xsdx=13s3
sdxx=sacos1|ax|
dxs=dxx2a2=ln|x+sa|
ln|x+sa|=sgn(x)cosh1|xa|=12ln(x+sxs), burada cosh1|xa|'nın pozitif değeri alınmalıdır.
xdxs=s
xdxs3=1s
xdxs5=13s3
xdxs7=15s5
xdxs2n+1=1(2n1)s2n1
x2mdxs2n+1=12n1x2m1s2n1+2m12n1x2m2dxs2n1
x2dxs=xs2+a22ln|x+sa|
x2dxs3=xs+ln|x+sa|
x4dxs=x3s4+38a2xs+38a4ln|x+sa|
x4dxs3=xs2a2xs+32a2ln|x+sa|
x4dxs5=xs13x3s3+ln|x+sa|
x2mdxs2n+1=(1)nm1a2(nm)i=0nm112(m+i)+1(nm1i)x2(m+i)+1s2(m+i)+1(n>m0)
dxs3=1a2xs
dxs5=1a4[xs13x3s3]
dxs7=1a6[xs23x3s3+15x5s5]
dxs9=1a8[xs33x3s3+35x5s517x7s7]
x2dxs5=1a2x33s3
x2dxs7=1a4[13x3s315x5s5]
x2dxs9=1a6[13x3s325x5s5+17x7s7]

t=a2x2 içeren integraller

tdx=12(xt+a2arcsinxa)(|x||a|)
xtdx=13t3(|x||a|)
tdxx=taln|a+tx|(|x||a|)
dxt=arcsinxa(|x||a|)
x2dxt=12(xt+a2arcsinxa)(|x||a|)
tdx=12(xtsgnxcosh1|xa|)(for|x||a|)

R=ax2+bx+c içeren integraller

dxR=1aln|2aR+2ax+b|(for a>0)
dxR=1asinh12ax+b4acb2(fora>04acb2>0)
dxR=1aln|2ax+b|(for a>04acb2=0)
dxR=1aarcsin2ax+bb24ac(fora<04acb2<0|2ax+b|<b24ac)
dxR3=4ax+2b(4acb2)R
dxR5=4ax+2b3(4acb2)R(1R2+8a4acb2)
dxR2n+1=2(2n1)(4acb2)(2ax+bR2n1+4a(n1)dxR2n1)
xRdx=Rab2adxR
xR3dx=2bx+4c(4acb2)R
xR2n+1dx=1(2n1)aR2n1b2adxR2n+1
dxxR=1cln(2cR+bx+2cx)
dxxR=1csinh1(bx+2c|x|4acb2)

S=ax+b içeren integraller

dxxax+b=2btanh1ax+bb
ax+bxdx=2(ax+bbtanh1ax+bb)
xnax+bdx=2a(2n+1)(xnax+bbnxn1ax+bdx)
xnax+bdx=22n+1(xn+1ax+b+bxnax+bnbxn1ax+bdx)

Kaynakça

Ayrıca bakınız

Şablon:İntegrallerin listeleri