What is the integral from 1 to e^9 of 1/(x(1+ln(x))) ?

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

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integral from 1 to e^9 of 1/(x(1+ln(x)))

\int_{1}^{e^{9}} \frac{1}{x ( 1 + ln ( x ) )} d x

ln (10)

Decimal

2.30258 \dots

Solution steps

\int_{1}^{e^{9}} \frac{1}{x ( 1 + ln ( x ) )} d x

Apply u-substitution

: \int_{1}^{10} \frac{1}{u} d u

= \int_{1}^{10} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= [ln ∣ u ∣]_{1}^{10}

Compute the boundaries:

ln (10)

= ln (10)

\int_{- 1}^{0} (2 x + x^{2} - x^{3}) d x

\int_{0}^{1} 1 0^{3 x} + 1 0^{- 3 x} d x

\int_{0}^{10} ln (x) d x

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

\int_{0}^{1} 0.2 + c x d x

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