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


## r = √(x^2+a^2) içeren integraller.
 ∫r dx = 1/2(x r +a^2 ln(x+r))
 ∫r^3 dx = 1/4xr^3+1/83a^2xr+3/8a^4ln(x+r)
 ∫r^5 dx = 1/6xr^5+5/24a^2xr^3+5/16a^4xr+5/16a^6ln(x+r)
 ∫x r dx=r^3/3
 ∫x r^3 dx=r^5/5
 ∫x r^2n+1 dx=r^2n+3/2n+3 
 ∫x^2 r dx= xr^3/4-a^2xr/8-a^4/8ln(x+r)
 ∫x^2 r^3 dx= xr^5/6-a^2xr^3/24-a^4xr/16-a^6/16ln(x+r)
 ∫x^3 r dx = r^5/5 - a^2 r^3/3
 ∫x^3 r^3 dx = r^7/7-a^2r^5/5 
 ∫x^3 r^2n+1 dx = r^2n+5/2n+5 - a^3 r^2n+3/2n+3
 ∫x^4 r dx= x^3r^3/6-a^2xr^3/8+a^4xr/16+a^6/16ln(x+r)
 ∫x^4 r^3 dx= x^3r^5/8-a^2xr^5/16+a^4xr^3/64+3a^6xr/128+3a^8/128ln(x+r)
 ∫x^5 r dx = r^7/7 - 2 a^2 r^5/5 + a^4 r^3/3
 ∫x^5 r^3 dx = r^9/9 - 2 a^2 r^7/7 + a^4 r^5/5
 ∫x^5 r^2n+1 dx = r^2n+7/2n+7 - 2a^2r^2n+5/2n+5+a^4 r^2n+3/2n+3 
 ∫r dx/x = r-aln|a+r/x| = r - a sinh^-1a/x
 ∫r^3 dx/x = r^3/3+a^2r-a^3ln|a+r/x|
 ∫r^5 dx/x = r^5/5+a^2r^3/3+a^4r-a^5ln|a+r/x|
 ∫r^7 dx/x = r^7/7+a^2r^5/5+a^4r^3/3+a^6r-a^7ln|a+r/x|
 ∫dx/r = sinh^-1x/a = ln|x+r|
 ∫dx/r^3 = x/a^2r
 ∫x dx/r = r
 ∫x dx/r^3 = -1/r
 ∫x^2 dx/r = x/2r-a^2/2 sinh^-1x/a = x/2r-a^2/2ln|x+r|
 ∫dx/xr = -1/a sinh^-1a/x = -1/aln|a+r/x|


## s = √(x^2-a^2) içeren integraller.
(x^2 gt;a^2) olduğunu varsayın. (x^2 lt;a^2) için bir sonraki bölüme bakınız.
 ∫xs dx = 1/3s^3
 ∫s dx/x = s - acos^-1|a/x|
 ∫dx/s = ∫dx/√(x^2-a^2) =ln|x+s/a|
 ln|x+s/a|
=sgn(x)cosh^-1|x/a|
=1/2ln(x+s/x-s), burada cosh^-1|x/a|'nın pozitif değeri alınmalıdır.
 ∫x dx/s = s
 ∫x dx/s^3 = -1/s
 ∫x dx/s^5 = -1/3s^3
 ∫x dx/s^7 = -1/5s^5
 ∫x dx/s^2n+1 = -1/(2n-1)s^2n-1 
 ∫x^2m dx/s^2n+1
= -1/2n-1x^2m-1/s^2n-1+2m-1/2n-1∫x^2m-2 dx/s^2n-1
 ∫x^2 dx/s
= xs/2+a^2/2ln|x+s/a|
 ∫x^2 dx/s^3
= -x/s+ln|x+s/a|
 ∫x^4 dx/s
= x^3s/4+3/8a^2xs+3/8a^4ln|x+s/a| 
 ∫x^4 dx/s^3
= xs/2-a^2x/s+3/2a^2ln|x+s/a| 
 ∫x^4 dx/s^5
= -x/s-1/3x^3/s^3+ln|x+s/a| 
 ∫x^2m dx/s^2n+1
= (-1)^n-m1/a^2(n-m)∑_i=0^n-m-11/2(m+i)+1n-m-1 ix^2(m+i)+1/s^2(m+i)+1 n gt;m≥0
 ∫dx/s^3=-1/a^2x/s
 ∫dx/s^5=1/a^4[x/s-1/3x^3/s^3]
 ∫dx/s^7
=-1/a^6[x/s-2/3x^3/s^3+1/5x^5/s^5]
 ∫dx/s^9
=1/a^8[x/s-3/3x^3/s^3+3/5x^5/s^5-1/7x^7/s^7]
 ∫x^2 dx/s^5=-1/a^2x^3/3s^3
 ∫x^2 dx/s^7
= 1/a^4[1/3x^3/s^3-1/5x^5/s^5]
 ∫x^2 dx/s^9
= -1/a^6[1/3x^3/s^3-2/5x^5/s^5+1/7x^7/s^7]


## t = √(a^2-x^2) içeren integraller.
 ∫t dx = 1/2(xt+a^2arcsinx/a) |x|≤|a|
 ∫xt dx = -1/3 t^3 |x|≤|a|
 ∫t dx/x = t-aln|a+t/x| |x|≤|a|
 ∫dx/t = arcsinx/a |x|≤|a|
 ∫x^2 dx/t = 1/2(-xt+a^2arcsinx/a) |x|≤|a|
 ∫t dx = 1/2(xt-x cosh^-1|x/a|) |x|≥|a|


## R = √(ax^2+bx+c) içeren integraller.
 ∫dx/R = 1/√(a)ln|2√(a)R+2ax+b| a gt;0
 ∫dx/R = 1/√(a) sinh^-12ax+b/√(4ac-b^2) a gt;04ac-b^2 gt;0
 ∫dx/R = 1/√(a)ln|2ax+b| a gt;04ac-b^2=0
 ∫dx/R = -1/√(-a)arcsin2ax+b/√(b^2-4ac) a lt;04ac-b^2 lt;0|2ax+b| lt;√(b^2-4ac)
 ∫dx/R^3 = 4ax+2b/(4ac-b^2)R
 ∫dx/R^5 = 4ax+2b/3(4ac-b^2)R(1/R^2+8a/4ac-b^2)
 ∫dx/R^2n+1 = 2/(2n-1)(4ac-b^2)(2ax+b/R^2n-1+4a(n-1)∫dx/R^2n-1)
 ∫x/R dx = R/a-b/2a∫dx/R
 ∫x/R^3 dx = -2bx+4c/(4ac-b^2)R
 ∫x/R^2n+1 dx = -1/(2n-1)aR^2n-1-b/2a∫dx/R^2n+1
 ∫dx/xR=-1/√(c)ln(2√(c)R+bx+2c/x)
 ∫dx/xR=-1/√(c)sinh^-1(bx+2c/|x|√(4ac-b^2))


## S = √(ax+b) içeren integraller.
 ∫dx/x√(ax + b) = -2/√(b)tanh^-1√(ax + b/b) 
 ∫√(ax + b)/x dx = 2(√(ax + b) - √(b)tanh^-1√(ax + b/b)) 
 ∫x^n/√(ax + b) dx = 2/a(2n+1)
 ∫x^n √(ax + b) dx = 2/2n +1(x^n+1 √(ax + b) + bx^n √(ax + b) - nb∫x^n-1√(ax + b) dx )