What is the integral from 0 to 28 of x/(sqrt(25+2x)) ?

The integral from 0 to 28 of x/(sqrt(25+2x)) is 152/3

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integral from 0 to 28 of x/(sqrt(25+2x))

\int_{0}^{28} \frac{x}{25 + 2 x} d x

\frac{152}{3}

Solution steps

\int_{0}^{28} \frac{x}{25 + 2 x} d x

Apply u-substitution

: \int_{5}^{9} \frac{u ^{2} - 25}{2} d u

= \int_{5}^{9} \frac{u ^{2} - 25}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int_{5}^{9} u^{2} - 25 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

= \frac{1}{2} (\int_{5}^{9} u^{2} d u - \int_{5}^{9} 25 d u)

\int_{5}^{9} u^{2} d u = \frac{604}{3}

\int_{5}^{9} 25 d u = 100

= \frac{1}{2} (\frac{604}{3} - 100)

Simplify

\frac{1}{2} (\frac{604}{3} - 100) : \frac{152}{3}

= \frac{152}{3}

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What is the integral from 0 to 28 of x/(sqrt(25+2x)) ?
The integral from 0 to 28 of x/(sqrt(25+2x)) is 152/3