ⓘ लघुगणकीय फलनों के समाकलों की सूची. नीचे लघुगणकीय फलनों के समाकलों की सूची दी गयी है। ध्यान दें: इस पूरे लेख में यह माना गया है कि x > 0। इसके अलावा समाकलन स् ..

                                     

ⓘ लघुगणकीय फलनों के समाकलों की सूची

नीचे लघुगणकीय फलनों के समाकलों की सूची दी गयी है।

ध्यान दें: इस पूरे लेख में यह माना गया है कि x > 0। इसके अलावा समाकलन स्थिरांक को लिखने के बजाय छोड दिया गया है।

                                     

1. केवल लघुगणकीय फलन वाले समाकल

∫ log a ⁡ x d x = x ln ⁡ x − x ln ⁡ a {\displaystyle \int \log _{a}x\,dx={\frac {x\ln x-x}{\ln a}}} ∫ ln ⁡ a x d x = x ln ⁡ a x − x {\displaystyle \int \lnax\,dx=x\lnax-x} ∫ ln ⁡ a x + b d x = a x + b ln ⁡ a x + b − a x a {\displaystyle \int \lnax+b\,dx={\frac {ax+b\lnax+b-ax}{a}}} ∫ ln ⁡ x 2 d x = x ln ⁡ x 2 − 2 x ln ⁡ x + 2 x {\displaystyle \int \ln x^{2}\,dx=x\ln x^{2}-2x\ln x+2x} ∫ ln ⁡ x n d x = x ∑ k = 0 n − 1 n − k n! k! ln ⁡ x k {\displaystyle \int \ln x^{n}\,dx=x\sum _{k=0}^{n}-1^{n-k}{\frac {n!}{k!}}\ln x^{k}} ∫ d x ln ⁡ x = ln ⁡ | ln ⁡ x | + ln ⁡ x + ∑ k = 2 ∞ ln ⁡ x k ⋅ k! {\displaystyle \int {\frac {dx}{\ln x}}=\ln |\ln x|+\ln x+\sum _{k=2}^{\infty }{\frac {\ln x^{k}}{k\cdot k!}}} ∫ d x ln ⁡ x = li ⁡ x {\displaystyle \int {\frac {dx}{\ln x}}=\operatorname {li} x}, the logarithmic integral. ∫ d x ln ⁡ x n = − x n − 1 ln ⁡ x n − 1 + 1 n − 1 ∫ d x ln ⁡ x n − 1 for n ≠ 1 {\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{}}}
                                     

2. लघुगणक तथा घात वाले फलनों के समाकलन

∫ x m ln ⁡ x d x = x m + 1 ln ⁡ x m + 1 − 1 m + 1 2) for m ≠ − 1 {\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{}}} ∫ x m ln ⁡ x n d x = x m + 1 ln ⁡ x n m + 1 − n m + 1 ∫ x m ln ⁡ x n − 1 d x for m ≠ − 1 {\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{}}} ∫ ln ⁡ x n d x = ln ⁡ x n + 1 n + 1 for n ≠ − 1 {\displaystyle \int {\frac {\ln x^{n}\,dx}{x}}={\frac {\ln x^{n+1}}{n+1}}\qquad {\mbox{for }}n\neq -1{\mbox{}}} ∫ ln ⁡ x n d x = ln ⁡ x n 2 n for n ≠ 0 {\displaystyle \int {\frac {\ln {x^{n}}\,dx}{x}}={\frac {\ln {x^{n}}^{2}}{2n}}\qquad {\mbox{for }}n\neq 0{\mbox{}}} ∫ ln ⁡ x d x m = − ln ⁡ x m − 1 x m − 1 − 1 m − 1 2 x m − 1 for m ≠ 1 {\displaystyle \int {\frac {\ln x\,dx}{x^{m}}}=-{\frac {\ln x}{m-1x^{m-1}}}-{\frac {1}{m-1^{2}x^{m-1}}}\qquad {\mbox{for }}m\neq 1{\mbox{}}} ∫ ln ⁡ x n d x m = − ln ⁡ x n m − 1 x m − 1 + n m − 1 ∫ ln ⁡ x n − 1 d x m for m ≠ 1 {\displaystyle \int {\frac {\ln x^{n}\,dx}{x^{m}}}=-{\frac {\ln x^{n}}{m-1x^{m-1}}}+{\frac {n}{m-1}}\int {\frac {\ln x^{n-1}dx}{x^{m}}}\qquad {\mbox{for }}m\neq 1{\mbox{}}} ∫ x m d x ln ⁡ x n = − x m + 1 n − 1 ln ⁡ x n − 1 + m + 1 n − 1 ∫ x m d x ln ⁡ x n − 1 for n ≠ 1 {\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{}}} ∫ d x ln ⁡ x = ln ⁡ | ln ⁡ x | {\displaystyle \int {\frac {dx}{x\ln x}}=\ln \left|\ln x\right|} ∫ d x ln ⁡ x ln ⁡ ln ⁡ x = ln ⁡ | ln ⁡ | ln ⁡ x | | {\displaystyle \int {\frac {dx}{x\ln x\\ln x}}=\ln \left|\ln \left|\ln x\right|\right|}, etc. ∫ d x ln ⁡ ln ⁡ x = li ⁡ ln ⁡ x {\displaystyle \int {\frac {dx}{x\\ln x}}=\operatorname {li} \ln x} ∫ d x ln ⁡ x = ln ⁡ | ln ⁡ x | + ∑ k = 1 ∞ − 1 k n − 1 k ln ⁡ x k ⋅ k! {\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!}}} ∫ d x ln ⁡ x n = − 1 n − 1 ln ⁡ x n − 1 for n ≠ 1 {\displaystyle \int {\frac {dx}{x\ln x^{n}}}=-{\frac {1}{n-1\ln x^{n-1}}}\qquad {\mbox{for }}n\neq 1{\mbox{}}} ∫ ln ⁡ x 2 + a 2 d x = x ln ⁡ x 2 + a 2 − 2 x + 2 a tan − 1 ⁡ x a {\displaystyle \int \lnx^{2}+a^{2}\,dx=x\lnx^{2}+a^{2}-2x+2a\tan ^{-1}{\frac {x}{a}}} ∫ x 2 + a 2 ln ⁡ x 2 + a 2 d x = 1 4 ln 2 ⁡ x 2 + a 2 {\displaystyle \int {\frac {x}{x^{2}+a^{2}}}\lnx^{2}+a^{2}\,dx={\frac {1}{4}}\ln ^{2}x^{2}+a^{2}}
                                     

3. लघुगणकीय तथा त्रिकोणमितीय फलनों से युक्त फलनों के समाकलन

∫ sin ⁡ ln ⁡ x d x = x 2 sin ⁡ ln ⁡ x − cos ⁡ ln ⁡ x) {\displaystyle \int \sin\ln x\,dx={\frac {x}{2}}\sin\ln x-\cos\ln x)} ∫ cos ⁡ ln ⁡ x d x = x 2 sin ⁡ ln ⁡ x + cos ⁡ ln ⁡ x) {\displaystyle \int \cos\ln x\,dx={\frac {x}{2}}\sin\ln x+\cos\ln x)}
                                     

4. Integrals involving logarithmic and exponential functions

∫ e x ln ⁡ x − x − 1 x d x = e x ln ⁡ x − x − ln ⁡ x {\displaystyle \int e^{x}\leftx\ln x-x-{\frac {1}{x}}\right\,dx=e^{x}x\ln x-x-\ln x} ∫ 1 e x 1 x − ln ⁡ x d x = ln ⁡ x e x {\displaystyle \int {\frac {1}{e^{x}}}\left{\frac {1}{x}}-\ln x\right\,dx={\frac {\ln x}{e^{x}}}} ∫ e x 1 ln ⁡ x − 1 x ln ⁡ x 2) d x = e x ln ⁡ x {\displaystyle \int e^{x}\left{\frac {1}{\ln x}}-{\frac {1}{x\ln x^{2}}}\right)\,dx={\frac {e^{x}}{\ln x}}}
                                     

5. n क्रमागत समाकल

n {\displaystyle n} क्रमागत समाकल consecutive integrations के लिए निम्नलिखित सूत्र का प्रयोग करने पर

∫ ln ⁡ x d x = x ln ⁡ x − 1 + C 0 {\displaystyle \int \ln x\,dx=x\ln x-1+C_{0}}

निम्नलिखित सामान्यीकरण प्राप्त होता है-

∫ ⋯ ∫ ln ⁡ x d x ⋯ d x = x n n! ln x − ∑ k = 1 n 1 k + ∑ k = 0 n − 1 C k x k k! {\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!}}}

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