What is the integral of 1/(x(ln(x))^p) ?

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

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integral of 1/(x(ln(x))^p)

\int \frac{1}{x ( ln ( x ) ) ^{p}} d x

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

Solution steps

\int \frac{1}{x ( ln ( x ) ) ^{p}} d x

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{( ln ( x ) ) ^{p} x} = (ln (x))^{- p} \frac{1}{x}

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

Simplify

(ln (x))^{- p} \frac{1}{x} : \frac{ln ^{- p} ( x )}{x}

= \int \frac{ln ^{- p} ( x )}{x} d x

Apply u-substitution

: \int u^{- p} d u

= \int u^{- p} d u

Apply the Power Rule

: \frac{u ^{- p + 1}}{- p + 1}

= \frac{u ^{- p + 1}}{- p + 1}

Substitute back

u = ln (x)

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

Add a constant to the solution

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

\frac{d}{d y} (c y^{- 6})

x \to 0 lim (cos (x) - sin (x))

\frac{d}{d x} (\frac{e ^{x}}{5 x})

\int - 3 arcsin (10 x) d x

(- 5, - 2), (5, 4)

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