List of integrals of logarithmic functions

The following is a list of integrals (antiderivative functions) of logarithmic functions. For a complete list of integral functions, please see table of integrals and list of integrals.

Related Topics:
Integral - Antiderivative - Logarithmic function - Table of integrals - List of integrals

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Note: x>0 is assumed throughout this article.

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: intln cx,dx = xln cx - x

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: int (ln x)^2; dx = x(ln x)^2 - 2xln x + 2x

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: int (ln cx)^n; dx = x(ln cx)^n - nint (ln cx)^{n-1} dx

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: int rac{dx}{ln x} = ln|ln x| + ln x + sum^infty_{i=2}rac{(ln x)^i}{icdot i!}

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: int rac{dx}{(ln x)^n} = -rac{x}{(n-1)(ln x)^{n-1}} + rac{1}{n-1}intrac{dx}{(ln x)^{n-1}} qquadmbox{(for }n eq 1mbox{)}

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: int x^mln x;dx = x^{m+1}left(rac{ln x}{m+1}-rac{1}{(m+1)^2} ight) qquadmbox{(for }m eq -1mbox{)}

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: int x^m (ln x)^n; dx = rac{x^{m+1}(ln x)^n}{m+1} - rac{n}{m+1}int x^m (ln x)^{n-1} dx qquadmbox{(for }m eq -1mbox{)}

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: int rac{(ln x)^n; dx}{x} = rac{(ln x)^{n+1}}{n+1} qquadmbox{(for }n eq -1mbox{)}

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: int rac{ln x,dx}{x^m} = -rac{ln x}{(m-1)x^{m-1}}-rac{1}{(m-1)^2 x^{m-1}} qquadmbox{(for }m eq 1mbox{)}

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: int rac{(ln x)^n; dx}{x^m} = -rac{(ln x)^n}{(m-1)x^{m-1}} + rac{n}{m-1}intrac{(ln x)^{n-1} dx}{x^m} qquadmbox{(for }m eq 1mbox{)}

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: int rac{x^m; dx}{(ln x)^n} = -rac{x^{m+1}}{(n-1)(ln x)^{n-1}} + rac{m+1}{n-1}intrac{x^m dx}{(ln x)^{n-1}} qquadmbox{(for }n eq 1mbox{)}

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: int rac{dx}{xln x} = ln|ln x|

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: int rac{dx}{x^nln x} = ln|ln x| + sum^infty_{i=1} (-1)^irac{(n-1)^i(ln x)^i}{icdot i!}

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: int rac{dx}{x (ln x)^n} = -rac{1}{(n-1)(ln x)^{n-1}} qquadmbox{(for }n eq 1mbox{)}

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: int sin (ln x);dx = rac{x}{2}(sin (ln x) - cos (ln x))

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: int cos (ln x);dx = rac{x}{2}(sin (ln x) + cos (ln x))

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