What is the integral from 2 to 3 of 1/(x-sqrt(x)) ?

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

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integral from 2 to 3 of 1/(x-sqrt(x))

\int_{2}^{3} \frac{1}{x - x} d x

ln ((4 - 23) (3 + 22))

Decimal

1.13893 \dots

Solution steps

\int_{2}^{3} \frac{1}{x - x} d x

Apply u-substitution

: \int_{2}^{3} \frac{2}{u - 1} d u

= \int_{2}^{3} \frac{2}{u - 1} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int_{2}^{3} \frac{1}{u - 1} d u

Apply u-substitution

: \int_{2 - 1}^{3 - 1} \frac{1}{v} d v

= 2 \cdot \int_{2 - 1}^{3 - 1} \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 2 [ln ∣ v ∣]_{2 - 1}^{3 - 1}

Compute the boundaries:

ln (3 - 1) - ln (2 - 1)

= 2 (ln (3 - 1) - ln (2 - 1))

Simplify

= ln ((4 - 23) (3 + 22))

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