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

The integral from 2 to 3 of x/(sqrt(x-2)) is 14/3

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

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

\frac{14}{3}

Decimal Notation

4.66666 \dots

Solution steps

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

Apply u-substitution

: \int_{0}^{1} 2 (u^{2} + 2) d u

= \int_{0}^{1} 2 (u^{2} + 2) d u

Take the constant out:

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

= 2 \cdot \int_{0}^{1} u^{2} + 2 d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int_{0}^{1} u^{2} d u + \int_{0}^{1} 2 d u)

\int_{0}^{1} u^{2} d u = \frac{1}{3}

\int_{0}^{1} 2 d u = 2

= 2 (\frac{1}{3} + 2)

Simplify

2 (\frac{1}{3} + 2) : \frac{14}{3}

= \frac{14}{3}

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What is the integral from 2 to 3 of x/(sqrt(x-2)) ?
The integral from 2 to 3 of x/(sqrt(x-2)) is 14/3