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

The integral from 1 to 4 of (x+1)/(2sqrt(x)) is 10/3

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integral from 1 to 4 of (x+1)/(2sqrt(x))

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

\frac{10}{3}

Decimal

3.33333 \dots

Solution steps

\int_{1}^{4} \frac{x + 1}{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_{1}^{4} \frac{x + 1}{x} d x

Expand

\frac{x + 1}{x} : x + \frac{1}{x}

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

\int_{1}^{4} x d x = \frac{14}{3}

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

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

Simplify

\frac{1}{2} (\frac{14}{3} + 2) : \frac{10}{3}

= \frac{10}{3}

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What is the integral from 1 to 4 of (x+1)/(2sqrt(x)) ?
The integral from 1 to 4 of (x+1)/(2sqrt(x)) is 10/3