লগ্যারিদমিক অপেক্ষকের সমাকলের তালিকা নিচে প্রদত্ত হল।

*এই নিবন্ধে* *x* > 0 হিসাবে গণ্য করা হয়েছে। এছাড়া সমাকলন স্থিরাঙ্ক বাদ দেওয়া হয়েছে।

লগ্যারিদমিক অপেক্ষক সমন্বিত সমাকল
---------------------------------

{\displaystyle \int \log \_{a}x\,dx={\frac {x\ln x-x}{\ln a}}}
{\displaystyle \int \ln(ax)\,dx=x\ln(ax)-x}
{\displaystyle \int \ln(ax+b)\,dx={\frac {(ax+b)\ln(ax+b)-ax}{a}}}
{\displaystyle \int (\ln x)^{2}\,dx=x(\ln x)^{2}-2x\ln x+2x}
{\displaystyle \int (\ln x)^{n}\,dx=x\sum \_{k=0}^{n}(-1)^{n-k}{\frac {n!}{k!}}(\ln x)^{k}}
{\displaystyle \int {\frac {dx}{\ln x}}=\ln |\ln x|+\ln x+\sum \_{k=2}^{\infty }{\frac {(\ln x)^{k}}{k\cdot k!}}}
{\displaystyle \int {\frac {dx}{\ln x}}=\operatorname {li} (x)}, লগ্যারিদমিক সমাকল
{\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{)}}}

লগ্যারিদমিক তথা সূচকীয় অপেক্ষক সমন্বিত সমাকল
---------------------------------------------

{\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{)}}}
{\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{)}}}
{\displaystyle \int {\frac {(\ln x)^{n}\,dx}{x}}={\frac {(\ln x)^{n+1}}{n+1}}\qquad {\mbox{(for }}n\neq -1{\mbox{)}}}
{\displaystyle \int {\frac {\ln {x^{n}}\,dx}{x}}={\frac {(\ln {x^{n}})^{2}}{2n}}\qquad {\mbox{(for }}n\neq 0{\mbox{)}}}
{\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{)}}}
{\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{)}}}
{\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{)}}}
{\displaystyle \int {\frac {dx}{x\ln x}}=\ln \left|\ln x\right|}
{\displaystyle \int {\frac {dx}{x\ln x\ln \ln x}}=\ln \left|\ln \left|\ln x\right|\right|}, etc.
{\displaystyle \int {\frac {dx}{x\ln \ln x}}=\operatorname {li} (\ln x)}
{\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!}}}
{\displaystyle \int {\frac {dx}{x(\ln x)^{n}}}=-{\frac {1}{(n-1)(\ln x)^{n-1}}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
{\displaystyle \int \ln(x^{2}+a^{2})\,dx=x\ln(x^{2}+a^{2})-2x+2a\tan ^{-1}{\frac {x}{a}}}
{\displaystyle \int {\frac {x}{x^{2}+a^{2}}}\ln(x^{2}+a^{2})\,dx={\frac {1}{4}}\ln ^{2}(x^{2}+a^{2})}

লগ্যারিদমিক তথা ত্রিকোণমিতিক অপেক্ষক সমন্বিত অপেক্ষকের সমাকল
------------------------------------------------------------

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

লগ্যারিদমিক এবং সূচকীয় অপেক্ষক সমন্বিত সমাকল
---------------------------------------------

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

*n* ক্রমাগত সমাকল
-----------------

{\displaystyle n} ক্রমাগত সমাকল (consecutive integrations)-এর জন্য নিম্নলিখিত সূত্রের প্রয়োগ করলে

{\displaystyle \int \ln x\,dx=x(\ln x-1)+C\_{0}}
নিম্নলিখিত সামান্যীকরণ প্রাপ্ত হয় –

{\displaystyle \int \dotsi \int \ln x\,dx\dotsm dx={\frac {x^{n}}{n!}}\left(\ln \,x-\sum \_{k=1}^{n}{\frac {1}{k}}\right)+\sum \_{k=0}^{n-1}C\_{k}{\frac {x^{k}}{k!}}}

তথ্যসূত্র
---------

* Milton Abramowitz and Irene A. Stegun, *Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*, 1964. A few integrals are listed on page 69

বহিঃসংযোগ
---------