Popis integrala logaritamskih funkcija

Slijedi popis integrala (antiderivacija funkcija) logaritamskih funkcija. Za potpun popis integrala funkcija, pogledati tablica integrala i popis integrala.

Valja uočiti: u članku se pretpostavlja da vrijedi x>0.

[math]\displaystyle{ \int\ln cx\;dx = x\ln cx - x + C }[/math]

[math]\displaystyle{ \int\ln (ax + b)\;dx = x\ln (ax +b) - x + \frac{b}{a}\ln (ax + b) + C }[/math]

[math]\displaystyle{ \int (\ln x)^2\; dx = x(\ln x)^2 - 2x\ln x + 2x + C }[/math]

[math]\displaystyle{ \int (\ln cx)^n\; dx = x(\ln cx)^n - n\int (\ln cx)^{n-1} dx + C }[/math]

[math]\displaystyle{ \int \frac{dx}{\ln x} = \ln|\ln x| + \ln x + \sum^\infty_{i=2}\frac{(\ln x)^i}{i\cdot i!} + C }[/math]

[math]\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}} + C \qquad\mbox{(za }n\neq 1\mbox{)} }[/math]

[math]\displaystyle{ \int x^m\ln x\;dx = x^{m+1}\left(\frac{\ln x}{m+1}-\frac{1}{(m+1)^2}\right) + C \qquad\mbox{(za }m\neq -1\mbox{)} }[/math]

[math]\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 + C \qquad\mbox{(za }m\neq -1\mbox{)} }[/math]

[math]\displaystyle{ \int \frac{(\ln x)^n\; dx}{x} = \frac{(\ln x)^{n+1}}{n+1} + C \qquad\mbox{(za }n\neq -1\mbox{)} }[/math]

[math]\displaystyle{ \int \frac{\ln{x^n}\;dx}{x} = \frac{(\ln{x^n})^2}{2n} + C \qquad\mbox{(za }n\neq 0\mbox{)} }[/math]

[math]\displaystyle{ \int \frac{\ln x\,dx}{x^m} = -\frac{\ln x}{(m-1)x^{m-1}}-\frac{1}{(m-1)^2 x^{m-1}} + C \qquad\mbox{(za }m\neq 1\mbox{)} }[/math]

[math]\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} + C \qquad\mbox{(za }m\neq 1\mbox{)} }[/math]

[math]\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}} + C \qquad\mbox{(za }n\neq 1\mbox{)} }[/math]

[math]\displaystyle{ \int \frac{dx}{x\ln x} = \ln \left|\ln x\right| + C }[/math]

[math]\displaystyle{ \int \frac{dx}{x^n\ln x} = \ln \left|\ln x\right| + \sum^\infty_{i=1} (-1)^i\frac{(n-1)^i(\ln x)^i}{i\cdot i!} + C }[/math]

[math]\displaystyle{ \int \frac{dx}{x (\ln x)^n} = -\frac{1}{(n-1)(\ln x)^{n-1}} + C \qquad\mbox{(za }n\neq 1\mbox{)} }[/math]

[math]\displaystyle{ \int \ln(x^2+a^2)\; dx = x\ln(x^2+a^2)-2x+2a\tan^{-1} \frac{x}{a} + C }[/math]

[math]\displaystyle{ \int \frac{x}{x^2+a^2}\ln(x^2+a^2)\; dx = \frac{1}{4} \ln^2(x^2+a^2) + C }[/math]

[math]\displaystyle{ \int \sin (\ln x)\;dx = \frac{x}{2}(\sin (\ln x) - \cos (\ln x)) + C }[/math]

[math]\displaystyle{ \int \cos (\ln x)\;dx = \frac{x}{2}(\sin (\ln x) + \cos (\ln x)) + C }[/math]

[math]\displaystyle{ \int e^x \left(x \ln x - x - \frac{1}{x}\right)\;dx = e^x (x \ln x - x - \ln x) + C }[/math]

[math]\displaystyle{ \int \frac{1}{e^x} \left( \frac{1}{x}-\ln x \right)\;dx = \frac{\ln x}{e^x} + C }[/math]

[math]\displaystyle{ \int e^x \left( \frac{1}{\ln x}- \frac{1}{x\ln^2 x} \right)\;dx = \frac{e^x}{\ln x} + C }[/math]

Izvori

• Milton Abramowitz i Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1964. Nekolicina integrala je izlistana na stranici 69 ove klasične knjige.