What is the integral from 4 to 9 of (2+x)/(2sqrt(x)) ?

The integral from 4 to 9 of (2+x)/(2sqrt(x)) is 25/3

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integral from 4 to 9 of (2+x)/(2sqrt(x))

\int_{4}^{9} \frac{2 + x}{2 x} d x

\frac{25}{3}

Decimal

8.33333 \dots

Solution steps

\int_{4}^{9} \frac{2 + x}{2 x} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int_{4}^{9} \frac{2 + x}{x} d x

Expand

\frac{2 + x}{x} : \frac{2}{x} + x

= \frac{1}{2} \cdot \int_{4}^{9} \frac{2}{x} + x d x

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_{4}^{9} \frac{2}{x} d x + \int_{4}^{9} x d x)

\int_{4}^{9} \frac{2}{x} d x = 4

\int_{4}^{9} x d x = \frac{38}{3}

= \frac{1}{2} (4 + \frac{38}{3})

Simplify

\frac{1}{2} (4 + \frac{38}{3}) : \frac{25}{3}

= \frac{25}{3}

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What is the integral from 4 to 9 of (2+x)/(2sqrt(x)) ?
The integral from 4 to 9 of (2+x)/(2sqrt(x)) is 25/3