What is the integral of (ln(x))/(x^m) ?

The integral of (ln(x))/(x^m) is 1/(-m+1)x^{-m+1}ln(x)-(x^{-m+1})/((-m+1)^2)+C

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integral of (ln(x))/(x^m)

\int \frac{ln ( x )}{x ^{m}} d x

\frac{1}{- m + 1} x^{- m + 1} ln (x) - \frac{x ^{- m + 1}}{( - m + 1 ) ^{2}} + C

Solution steps

\int \frac{ln ( x )}{x ^{m}} d x

Apply Integration By Parts

: \frac{1}{- m + 1} x^{- m + 1} ln (x) - \int \frac{1}{x ^{m} ( - m + 1 )} d x

= \frac{1}{- m + 1} x^{- m + 1} ln (x) - \int \frac{1}{x ^{m} ( - m + 1 )} d x

\int \frac{1}{x ^{m} ( - m + 1 )} d x = \frac{x ^{- m + 1}}{( - m + 1 ) ^{2}}

= \frac{1}{- m + 1} x^{- m + 1} ln (x) - \frac{x ^{- m + 1}}{( - m + 1 ) ^{2}}

Add a constant to the solution

= \frac{1}{- m + 1} x^{- m + 1} ln (x) - \frac{x ^{- m + 1}}{( - m + 1 ) ^{2}} + C

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What is the integral of (ln(x))/(x^m) ?
The integral of (ln(x))/(x^m) is 1/(-m+1)x^{-m+1}ln(x)-(x^{-m+1})/((-m+1)^2)+C