[  ] / pk  [] dx=

${\displaystyle \int a^{cx}\;dx={\frac {1}{c\ln a}}a^{cx}\qquad {\mbox{(para }}a>0,{\mbox{ }}a\neq 1{\mbox{)}}}$ [  ] / pk  [] dx=

${\displaystyle \int xe^{cx}\;dx={\frac {e^{cx}}{c^{2}}}(cx-1)}$  [  ] / pk  [] dx=

${\displaystyle \int x^{2}e^{cx}\;dx=e^{cx}\left({\frac {x^{2}}{c}}-{\frac {2x}{c^{2}}}+{\frac {2}{c^{3}}}\right)}$ [  ] / pk  [] dx=

${\displaystyle \int x^{n}e^{cx}\;dx={\frac {1}{c}}x^{n}e^{cx}-{\frac {n}{c}}\int x^{n-1}e^{cx}dx}$ [  ] / pk  [] dx=

${\displaystyle \int {\frac {e^{cx}\;dx}{x}}=\ln |x|+\sum _{i=1}^{\infty }{\frac {(cx)^{i}}{i\cdot i!}}}$ [  ] / pk  [] dx=

${\displaystyle \int {\frac {e^{cx}\;dx}{x^{n}}}={\frac {1}{n-1}}\left(-{\frac {e^{cx}}{x^{n-1}}}+c\int {\frac {e^{cx}dx}{x^{n-1}}}\right)\qquad {\mbox{(para }}n\neq 1{\mbox{)}}}$  [  ] / pk  [] dx=

${\displaystyle \int e^{cx}\ln x\;dx={\frac {1}{c}}e^{cx}\ln |x|-\operatorname {Ei} \,(cx)}$

 [  ] / pk  [] dx=

${\displaystyle \int e^{cx}\sin bx\;dx={\frac {e^{cx}}{c^{2}+b^{2}}}(c\sin bx-b\cos bx)}$   [  ] / pk  [] dx=

${\displaystyle \int e^{cx}\cos bx\;dx={\frac {e^{cx}}{c^{2}+b^{2}}}(c\cos bx+b\sin bx)}$  [  ] / pk  [] dx=

${\displaystyle \int e^{cx}\sin ^{n}x\;dx={\frac {e^{cx}\sin ^{n-1}x}{c^{2}+n^{2}}}(c\sin x-n\cos x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\sin ^{n-2}x\;dx}$  [  ] / pk  [] dx=

${\displaystyle \int e^{cx}\cos ^{n}x\;dx={\frac {e^{cx}\cos ^{n-1}x}{c^{2}+n^{2}}}(c\cos x+n\sin x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\cos ^{n-2}x\;dx}$

 [  ] / pk  [] dx=

${\displaystyle \int xe^{cx^{2}}\;dx={\frac {1}{2c}}\;e^{cx^{2}}}$   [  ] / pk  [] dx=

${\displaystyle \int {1 \over \sigma {\sqrt {2\pi }}}\,e^{-{(x-\mu )^{2}/2\sigma ^{2}}}\;dx={\frac {1}{2\sigma }}(1+{\mbox{erf}}\,{\frac {x-\mu }{\sigma {\sqrt {2}}}})}$  

 [  ] / pk  [] dx=

${\displaystyle \int e^{x^{2}}\,dx=e^{x^{2}}\left(\sum _{r=1}^{n}{\frac {1}{2^{n}x^{2n-1}}}\right)+{\frac {2n-1}{2^{n}}}\int {\frac {e^{x^{2}}\;dx}{x^{2n}}}}$ 

 [  ] / pk  [] dx=

${\displaystyle \int _{-\infty }^{\infty }e^{-ax^{2}}\,dx={\sqrt {\pi \over a}}}$   

 [  ] / pk  [] dx=

  

 [  ] / pk  [] dx=

${\displaystyle \int \ln cx\;dx=x\ln cx-x}$   

 [  ] / pk  [] dx=

${\displaystyle \int \ln(ax+b)\;dx={\frac {(ax+b)\ln(ax+b)-(ax+b)}{a}}}$  

 [  ] / pk  [] dx=

${\displaystyle \int (\ln x)^{2}\;dx=x(\ln x)^{2}-2x\ln x+2x}$ 

 [  ] / pk  [] dx=

${\displaystyle \int (\ln x)^{n}\;dx=x\sum _{k=0}^{n}(-1)^{k}{\frac {n!(\ln x)^{n-k}}{(n-k)!}}}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{\ln x}}=\ln |\ln x|+\ln x+\sum _{k=2}^{\infty }{\frac {(\ln x)^{k}}{k\cdot k!}}}$  

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{(\ln x)^{n}}}=-{\frac {x}{(n-1)(\ln x)^{n-1}}}+{\frac {1}{n-1}}\int {\frac {dx}{(\ln x)^{n-1}}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}$

 [  ] / pk  [] dx=

${\displaystyle \int x^{m}\ln x\;dx=x^{m+1}\left({\frac {\ln x}{m+1}}-{\frac {1}{(m+1)^{2}}}\right)\qquad {\mbox{(for }}m\neq -1{\mbox{)}}}$  

 [  ] / pk  [] dx=

${\displaystyle \int x^{m}(\ln x)^{n}\;dx={\frac {x^{m+1}(\ln x)^{n}}{m+1}}-{\frac {n}{m+1}}\int x^{m}(\ln x)^{n-1}dx\qquad {\mbox{(for }}m\neq -1{\mbox{)}}}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {(\ln x)^{n}\;dx}{x}}={\frac {(\ln x)^{n+1}}{n+1}}\qquad {\mbox{(for }}n\neq -1{\mbox{)}}}$   

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {\ln {x^{n}}\;dx}{x}}={\frac {(\ln {x^{n}})^{2}}{2n}}\qquad {\mbox{(for }}n\neq 0{\mbox{)}}}$   

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {\ln x\,dx}{x^{m}}}=-{\frac {\ln x}{(m-1)x^{m-1}}}-{\frac {1}{(m-1)^{2}x^{m-1}}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}}$  

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {(\ln x)^{n}\;dx}{x^{m}}}=-{\frac {(\ln x)^{n}}{(m-1)x^{m-1}}}+{\frac {n}{m-1}}\int {\frac {(\ln x)^{n-1}dx}{x^{m}}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}}$ 

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x^{m}\;dx}{(\ln x)^{n}}}=-{\frac {x^{m+1}}{(n-1)(\ln x)^{n-1}}}+{\frac {m+1}{n-1}}\int {\frac {x^{m}dx}{(\ln x)^{n-1}}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}$  

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{x\ln x}}=\ln \left|\ln x\right|}$  

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{x^{n}\ln x}}=\ln \left|\ln x\right|+\sum _{k=1}^{\infty }(-1)^{k}{\frac {(n-1)^{k}(\ln x)^{k}}{k\cdot k!}}}$  

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{x(\ln x)^{n}}}=-{\frac {1}{(n-1)(\ln x)^{n-1}}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}$  

 [  ] / pk  [] dx=

${\displaystyle \int \ln(x^{2}+a^{2})\;dx=x\ln(x^{2}+a^{2})-2x+2a\tan ^{-1}{\frac {x}{a}}}$  

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x}{x^{2}+a^{2}}}\ln(x^{2}+a^{2})\;dx={\frac {1}{4}}\ln ^{2}(x^{2}+a^{2})}$  

 [  ] / pk  [] dx=

${\displaystyle \int \sin(\ln x)\;dx={\frac {x}{2}}(\sin(\ln x)-\cos(\ln x))}$ 

 [  ] / pk  [] dx=

${\displaystyle \int \cos(\ln x)\;dx={\frac {x}{2}}(\sin(\ln x)+\cos(\ln x))}$  

 [  ] / pk  [] dx=

${\displaystyle \int e^{x}\left(x\ln x-x-{\frac {1}{x}}\right)\;dx=e^{x}(x\ln x-x-\ln x)}$ 

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {1}{e^{x}}}\left({\frac {1}{x}}-\ln x\right)\;dx={\frac {\ln x}{e^{x}}}}$  

 [  ] / pk  [] dx=

${\displaystyle \int e^{x}\left({\frac {1}{\ln x}}-{\frac {1}{x\ln ^{2}x}}\right)\;dx={\frac {e^{x}}{\ln x}}}$  

 [  ] / pk  [] dx=

 [  ] / pk  [] dx=

${\displaystyle \int r^{3}\;dx={\frac {1}{4}}xr^{3}+{\frac {1}{8}}3a^{2}xr+{\frac {3}{8}}a^{4}\ln \left({\frac {x+r}{a}}\right)}$ 

 [  ] / pk  [] dx=

${\displaystyle \int r^{5}\;dx={\frac {1}{6}}xr^{5}+{\frac {5}{24}}a^{2}xr^{3}+{\frac {5}{16}}a^{4}xr+{\frac {5}{16}}a^{6}\ln \left({\frac {x+r}{a}}\right)}$ 

 [  ] / pk  [] dx=

${\displaystyle \int xr\;dx={\frac {r^{3}}{3}}}$ 

 [  ] / pk  [] dx=

${\displaystyle \int xr^{3}\;dx={\frac {r^{5}}{5}}}$  

 [  ] / pk  [] dx=

${\displaystyle \int xr^{2n+1}\;dx={\frac {r^{2n+3}}{2n+3}}}$  

 [  ] / pk  [] dx=

${\displaystyle \int x^{2}r\;dx={\frac {xr^{3}}{4}}-{\frac {a^{2}xr}{8}}-{\frac {a^{4}}{8}}\ln \left({\frac {x+r}{a}}\right)}$

 [  ] / pk  [] dx=

${\displaystyle \int x^{2}r^{3}\;dx={\frac {xr^{5}}{6}}-{\frac {a^{2}xr^{3}}{24}}-{\frac {a^{4}xr}{16}}-{\frac {a^{6}}{16}}\ln \left({\frac {x+r}{a}}\right)}$

 [  ] / pk  [] dx=

${\displaystyle \int x^{3}r\;dx={\frac {r^{5}}{5}}-{\frac {a^{2}r^{3}}{3}}}$ 

 [  ] / pk  [] dx=

${\displaystyle \int x^{3}r^{3}\;dx={\frac {r^{7}}{7}}-{\frac {a^{2}r^{5}}{5}}}$ 

 [  ] / pk  [] dx=

${\displaystyle \int x^{3}r^{2n+1}\;dx={\frac {r^{2n+5}}{2n+5}}-{\frac {a^{3}r^{2n+3}}{2n+3}}}$  

 [  ] / pk  [] dx=

${\displaystyle \int x^{4}r\;dx={\frac {x^{3}r^{3}}{6}}-{\frac {a^{2}xr^{3}}{8}}-{\frac {a^{4}xr}{16}}+{\frac {a^{6}}{16}}\ln \left({\frac {x+r}{a}}\right)}$ 

 [  ] / pk  [] dx=

${\displaystyle \int x^{4}r^{3}\;dx={\frac {x^{3}r^{5}}{8}}-{\frac {a^{2}xr^{5}}{16}}-{\frac {a^{4}xr^{3}}{64}}+{\frac {3a^{6}xr}{128}}+{\frac {3a^{8}}{128}}\ln \left({\frac {x+r}{a}}\right)}$

 [  ] / pk  [] dx=

${\displaystyle \int x^{5}r\;dx={\frac {r^{7}}{7}}-{\frac {2a^{2}r^{5}}{5}}+{\frac {a^{4}r^{3}}{3}}}$

 [  ] / pk  [] dx=

${\displaystyle \int x^{5}r^{3}\;dx={\frac {r^{9}}{9}}-{\frac {2a^{2}r^{7}}{7}}+{\frac {a^{4}r^{5}}{5}}}$

 [  ] / pk  [] dx=

${\displaystyle \int x^{5}r^{2n+1}\;dx={\frac {r^{2n+7}}{2n+7}}-{\frac {2a^{2}r^{2n+5}}{2n+5}}+{\frac {a^{4}r^{2n+3}}{2n+3}}}$ 

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {r\;dx}{x}}=r-a\ln \left|{\frac {a+r}{x}}\right|=r-a\sinh ^{-1}{\frac {a}{x}}}$ 

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {r^{3}\;dx}{x}}={\frac {r^{3}}{3}}+a^{2}r-a^{3}\ln \left|{\frac {a+r}{x}}\right|}$ 

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {r^{5}\;dx}{x}}={\frac {r^{5}}{5}}+{\frac {a^{3}r^{3}}{3}}+a^{4}r-a^{5}\ln \left|{\frac {a+r}{x}}\right|}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {r^{7}\;dx}{x}}={\frac {r^{7}}{7}}+{\frac {a^{2}r^{5}}{5}}+{\frac {a^{4}r^{3}}{3}}+a^{6}r-a^{7}\ln \left|{\frac {a+r}{x}}\right|}$ 

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{r}}=\sinh ^{-1}{\frac {x}{a}}=\ln \left|x+r\right|}$ 

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{r^{3}}}={\frac {x}{a^{2}r}}}$  

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x\,dx}{r}}=r}$  

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x^{2}\;dx}{r}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\,\sinh ^{-1}{\frac {a}{x}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\ln \left|x+r\right|}$ 

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{xr}}=-{\frac {1}{a}}\,\sinh ^{-1}{\frac {a}{x}}=-{\frac {1}{a}}\ln \left|{\frac {a+r}{x}}\right|}$

 [  ] / pk  [] dx=

${\displaystyle \int xs\;dx={\frac {1}{3}}s^{3}}$ 

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {s\;dx}{x}}=s-a\cos ^{-1}\left|{\frac {a}{x}}\right|}$ 

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{s}}=\int {\frac {dx}{\sqrt {x^{2}-a^{2}}}}=\ln \left|{\frac {x+s}{a}}\right|}$ 

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x\;dx}{s}}=s}$ 

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x\;dx}{s^{3}}}=-{\frac {1}{s}}}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x\;dx}{s^{5}}}=-{\frac {1}{3s^{3}}}}$ 

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x\;dx}{s^{7}}}=-{\frac {1}{5s^{5}}}}$ 

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x\;dx}{s^{2n+1}}}=-{\frac {1}{(2n-1)s^{2n-1}}}}$ 

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x^{2m}\;dx}{s^{2n+1}}}=-{\frac {1}{2n-1}}{\frac {x^{2m-1}}{s^{2n-1}}}+{\frac {2m-1}{2n-1}}\int {\frac {x^{2m-2}\;dx}{s^{2n-1}}}}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x^{2}\;dx}{s}}={\frac {xs}{2}}+{\frac {a^{2}}{2}}\ln \left|{\frac {x+s}{a}}\right|}$ 

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x^{2}\;dx}{s^{3}}}=-{\frac {x}{s}}+\ln \left|{\frac {x+s}{a}}\right|}$ 

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x^{4}\;dx}{s}}={\frac {x^{3}s}{4}}+{\frac {3}{8}}a^{2}xs+{\frac {3}{8}}a^{4}\ln \left|{\frac {x+s}{a}}\right|}$ 

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x^{4}\;dx}{s^{3}}}={\frac {xs}{2}}-{\frac {a^{2}x}{s}}+{\frac {3}{2}}\ln \left|{\frac {x+s}{a}}\right|}$ 

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x^{4}\;dx}{s^{5}}}=-{\frac {x}{s}}-{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}+\ln \left|{\frac {x+s}{a}}\right|}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x^{2m}\;dx}{s^{2n+1}}}=(-1)^{n-m}{\frac {1}{a^{2(n-m)}}}\sum _{i=0}^{n-m-1}{\frac {1}{2(m+i)+1}}{n-m-1 \choose i}{\frac {x^{2(m+i)+1}}{s^{2(m+i)+1}}}\qquad {\mbox{(}}n>m\geq 0{\mbox{)}}}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{s^{5}}}={\frac {1}{a^{4}}}\left[{\frac {x}{s}}-{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}\right]}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{s^{7}}}=-{\frac {1}{a^{6}}}\left[{\frac {x}{s}}-{\frac {2}{3}}{\frac {x^{3}}{s^{3}}}+{\frac {1}{5}}{\frac {x^{5}}{s^{5}}}\right]}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{s^{9}}}={\frac {1}{a^{8}}}\left[{\frac {x}{s}}-{\frac {3}{3}}{\frac {x^{3}}{s^{3}}}+{\frac {3}{5}}{\frac {x^{5}}{s^{5}}}-{\frac {1}{7}}{\frac {x^{7}}{s^{7}}}\right]}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x^{2}\;dx}{s^{5}}}=-{\frac {1}{a^{2}}}{\frac {x^{3}}{3s^{3}}}}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x^{2}\;dx}{s^{7}}}={\frac {1}{a^{4}}}\left[{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}-{\frac {1}{5}}{\frac {x^{5}}{s^{5}}}\right]}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x^{2}\;dx}{s^{9}}}=-{\frac {1}{a^{6}}}\left[{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}-{\frac {2}{5}}{\frac {x^{5}}{s^{5}}}+{\frac {1}{7}}{\frac {x^{7}}{s^{7}}}\right]}$

 [  ] / pk  [] dx=

${\displaystyle \int t\;dx={\frac {1}{2}}\left(xt+a^{2}\sin ^{-1}{\frac {x}{a}}\right)\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}$

 [  ] / pk  [] dx=

${\displaystyle \int xt\;dx=-{\frac {1}{3}}t^{3}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {t\;dx}{x}}=t-a\ln \left|{\frac {a+t}{x}}\right|\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{t}}=\sin ^{-1}{\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x^{2}\;dx}{t}}=-{\frac {x}{2}}t+{\frac {a^{2}}{2}}\sin ^{-1}{\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}$

 [  ] / pk  [] dx=

${\displaystyle \int t\;dx={\frac {1}{2}}\left(xt-\operatorname {sgn} x\,\cosh ^{-1}\left|{\frac {x}{a}}\right|\right)\qquad {\mbox{(for }}|x|\geq |a|{\mbox{)}}}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\ln \left|2{\sqrt {aR}}+2ax+b\right|\qquad {\mbox{(for }}a>0{\mbox{)}}}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\,\sinh ^{-1}{\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{(for }}a>0{\mbox{, }}4ac-b^{2}>0{\mbox{)}}}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\ln |2ax+b|\quad {\mbox{(for }}a>0{\mbox{, }}4ac-b^{2}=0{\mbox{)}}}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}=-{\frac {1}{\sqrt {-a}}}\arcsin {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\mbox{(for }}a<0{\mbox{, }}4ac-b^{2}<0{\mbox{)}}}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{\sqrt {(ax^{2}+bx+c)^{3}}}}={\frac {4ax+2b}{(4ac-b^{2}){\sqrt {R}}}}}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{\sqrt {(ax^{2}+bx+c)^{5}}}}={\frac {4ax+2b}{3(4ac-b^{2}){\sqrt {R}}}}\left({\frac {1}{R}}+{\frac {8a}{4ac-b^{2}}}\right)}$v

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{\sqrt {(ax^{2}+bx+c)^{2n+1}}}}={\frac {4ax+2b}{(2n-1)(4ac-b^{2})R^{(2n-1)/2}}}+{\frac {8a(n-1)}{(2n-1)(4ac-b^{2})}}\int {\frac {dx}{R^{(2n-1)/2}}}}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x\;dx}{\sqrt {(ax^{2}+bx+c)^{3}}}}=-{\frac {2bx+4c}{(4ac-b^{2}){\sqrt {R}}}}}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x\;dx}{\sqrt {(ax^{2}+bx+c)^{2n+1}}}}=-{\frac {1}{(2n-1)aR^{(2n-1)/2}}}-{\frac {b}{2a}}\int {\frac {dx}{R^{(2n+1)/2}}}}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{x{\sqrt {ax^{2}+bx+c}}}}=-{\frac {1}{\sqrt {c}}}\ln \left({\frac {2{\sqrt {cR}}+bx+2c}{x}}\right)}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{x{\sqrt {ax^{2}+bx+c}}}}=-{\frac {1}{\sqrt {c}}}\sinh ^{-1}\left({\frac {bx+2c}{|x|{\sqrt {4ac-b^{2}}}}}\right)}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {dx}{x{\sqrt {ax+b}}}}\,=\,{\frac {-2}{\sqrt {b}}}\tanh ^{-1}{\sqrt {\frac {ax+b}{b}}}}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {\sqrt {ax+b}}{x}}\,dx\;=\;2\left({\sqrt {ax+b}}-{\sqrt {b}}\tanh ^{-1}{\sqrt {\frac {ax+b}{b}}}\right)}$

 [  ] / pk  [] dx=

${\displaystyle \int {\frac {x^{n}}{\sqrt {ax+b}}}\,dx\;=\;{\frac {2}{a}}\left(x^{n+1}{\sqrt {ax+b}}+bx^{n}{\sqrt {ax+b}}-nb\int {\frac {x^{n-1}}{\sqrt {ax+b}}}\right)}$

 [  ] / pk  [] dx=

${\displaystyle \int x^{n}{\sqrt {ax+b}}\,dx\;=\;{\frac {2}{2n+1}}\left(x^{n+1}{\sqrt {ax+b}}+bx^{n}{\sqrt {ax+b}}-nb\int x^{n-1}{\sqrt {ax+b}}\,dx\right)}$

 [  ] / pk  [] dx=