Integrals with Logarithms

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Indefinite integrals of expressions containing logarithms.

\int ln a x d x = x ln a x - x + C

\int x ln x d x = \frac{1}{2} x^{2} ln x - \frac{x ^{2}}{4} + C

\int x^{2} ln x d x = \frac{1}{3} x^{3} ln x - \frac{x ^{3}}{9} + C

\int x^{n} ln x d x = x^{n + 1} (\frac{ln x}{n + 1} - \frac{1}{( n + 1 ) ^{2}}) + C, n \neq = - 1

\int \frac{ln a x}{x} d x = \frac{1}{2} (ln a x)^{2} + C

\int \frac{ln x}{x ^{2}} d x = - \frac{1}{x} - \frac{ln x}{x} + C

\int ln (a x + b) d x = (x + \frac{b}{a}) ln (a x + b) - x + C, a \neq = 0

\int ln (x^{2} + a^{2}) d x = x ln (x^{2} + a^{2}) + 2 a arctan \frac{x}{a} - 2 x + C

\int ln (x^{2} - a^{2}) d x = x ln (x^{2} - a^{2}) + a ln \frac{x + a}{x - a} - 2 x + C

\int ln (a x^{2} + b x + c) d x = \frac{1}{a} 4 a c - b^{2} arctan \frac{2 a x + b}{4 a c - b ^{2}} - 2 x + (\frac{b}{2 a} + x) ln (a x^{2} + b x + c) + C

\int x ln (a x + b) d x = \frac{b x}{2 a} - \frac{1}{4} x^{2} + \frac{1}{2} (x^{2} - \frac{b ^{2}}{a ^{2}}) ln (a x + b) + C

\int x ln (a^{2} - b^{2} x^{2}) d x = - \frac{1}{2} x^{2} + \frac{1}{2} (x^{2} - \frac{a ^{2}}{b ^{2}}) ln (a^{2} - b^{2} x^{2}) + C

\int (ln x)^{2} d x = 2 x - 2 x ln x + x (ln x)^{2} + C

\int (ln x)^{3} d x = - 6 x + x (ln x)^{3} - 3 x (ln x)^{2} + 6 x ln x + C

\int x (ln x)^{2} d x = \frac{x ^{2}}{4} + \frac{1}{2} x^{2} (ln x)^{2} - \frac{1}{2} x^{2} ln x + C

\int x^{2} (ln x)^{2} d x = \frac{2 x ^{3}}{27} + \frac{1}{3} x^{3} (ln x)^{2} - \frac{2}{9} x^{3} ln x + C