What is the integral from 1 to e of (ln(x))/(x^{10)} ?

The integral from 1 to e of (ln(x))/(x^{10)} is -(10)/(81e^9)+1/81

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integral from 1 to e of (ln(x))/(x^{10)}

\int_{1}^{e} \frac{ln ( x )}{x ^{10}} d x

- \frac{10}{81 e ^{9}} + \frac{1}{81}

Decimal Notation

0.01233 \dots

Solution steps

\int_{1}^{e} \frac{ln ( x )}{x ^{10}} d x

Apply Integration By Parts

: [\frac{- ln ( x ) - 9 x ^{9} \cdot \int - \frac{1}{9 x ^{10}} d x}{9 x ^{9}}]_{1}^{e}

= [\frac{- ln ( x ) - 9 x ^{9} \cdot \int - \frac{1}{9 x ^{10}} d x}{9 x ^{9}}]_{1}^{e}

\int - \frac{1}{9 x ^{10}} d x = \frac{1}{81 x ^{9}}

= [\frac{- ln ( x ) - 9 x ^{9} \frac{1}{81 x ^{9}}}{9 x ^{9}}]_{1}^{e}

\frac{- ln ( x ) - 9 x ^{9} \frac{1}{81 x ^{9}}}{9 x ^{9}} = \frac{- 9 ln ( x ) - 1}{81 x ^{9}}

= [\frac{- 9 ln ( x ) - 1}{81 x ^{9}}]_{1}^{e}

Compute the boundaries:

- \frac{10}{81 e ^{9}} + \frac{1}{81}

= - \frac{10}{81 e ^{9}} + \frac{1}{81}

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What is the integral from 1 to e of (ln(x))/(x^{10)} ?
The integral from 1 to e of (ln(x))/(x^{10)} is -(10)/(81e^9)+1/81