What is the integral from 1 to e of (ln(x))/(x^3) ?

The integral from 1 to e of (ln(x))/(x^3) is -3/(4e^2)+1/4

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integral from 1 to e of (ln(x))/(x^3)

\int_{1}^{e} \frac{ln ( x )}{x ^{3}} d x

- \frac{3}{4 e ^{2}} + \frac{1}{4}

Decimal

0.14849 \dots

Solution steps

\int_{1}^{e} \frac{ln ( x )}{x ^{3}} d x

Apply Integration By Parts

: [\frac{- ln ( x ) - 2 x ^{2} \cdot \int - \frac{1}{2 x ^{3}} d x}{2 x ^{2}}]_{1}^{e}

= [\frac{- ln ( x ) - 2 x ^{2} \cdot \int - \frac{1}{2 x ^{3}} d x}{2 x ^{2}}]_{1}^{e}

\int - \frac{1}{2 x ^{3}} d x = \frac{1}{4 x ^{2}}

= [\frac{- ln ( x ) - 2 x ^{2} \frac{1}{4 x ^{2}}}{2 x ^{2}}]_{1}^{e}

\frac{- ln ( x ) - 2 x ^{2} \frac{1}{4 x ^{2}}}{2 x ^{2}} = \frac{- 2 ln ( x ) - 1}{4 x ^{2}}

= [\frac{- 2 ln ( x ) - 1}{4 x ^{2}}]_{1}^{e}

Compute the boundaries:

- \frac{3}{4 e ^{2}} + \frac{1}{4}

= - \frac{3}{4 e ^{2}} + \frac{1}{4}

f (θ) = 3 tan (θ)

x \to + (- 3) lim (5)

\frac{d}{d x} (ln (sin (\frac{x}{2})))

\int (3^{5 x - 2}) d x

\int_{- 1}^{3} 3 x e^{2 x^{2} - 5} d x

What is the integral from 1 to e of (ln(x))/(x^3) ?
The integral from 1 to e of (ln(x))/(x^3) is -3/(4e^2)+1/4