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

The integral from 1 to 4 of (x+4)/(2sqrt(x)) is 19/3

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

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

\frac{19}{3}

Decimal

6.33333 \dots

Solution steps

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

Expand

\frac{x + 4}{x} : x + \frac{4}{x}

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

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

\int_{1}^{4} \frac{4}{x} d x = 8

= \frac{1}{2} (\frac{14}{3} + 8)

Simplify

\frac{1}{2} (\frac{14}{3} + 8) : \frac{19}{3}

= \frac{19}{3}

6 x (x^{2} + 1)^{2}

x^{^{'}}

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\int (sec (x) - tan (x))^{2} d x

What is the integral from 1 to 4 of (x+4)/(2sqrt(x)) ?
The integral from 1 to 4 of (x+4)/(2sqrt(x)) is 19/3