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

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

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

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

ln (3)

Decimal

1.09861 \dots

Solution steps

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

Apply u-substitution

: \int_{1}^{3} \frac{1}{u} d u

= \int_{1}^{3} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= [ln ∣ u ∣]_{1}^{3}

Compute the boundaries:

ln (3)

= ln (3)

\int_{- 2}^{- 1} \frac{1}{x x ^{2} - 1} d x

\int_{- 1}^{1} t^{2} d t

\int_{0}^{t} cos (x) d x

\int_{0}^{π} cos^{2} (\frac{x}{2}) d x

\int_{0}^{1} a x + b x^{2} d x

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