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

The integral of (x+1)/(2sqrt(x)) is 1/3 sqrt(x)(x+3)+C

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integral of (x+1)/(2sqrt(x))

\int \frac{x + 1}{2 x} d x

\frac{1}{3} x (x + 3) + C

Solution steps

\int \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 \frac{x + 1}{x} d x

Expand

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

= \frac{1}{2} \cdot \int 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 x d x + \int \frac{1}{x} d x)

\int x d x = \frac{2}{3} x^{\frac{3}{2}}

\int \frac{1}{x} d x = 2 x

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

Simplify

\frac{1}{2} (\frac{2}{3} x^{\frac{3}{2}} + 2 x) : \frac{1}{3} x (x + 3)

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

Add a constant to the solution

= \frac{1}{3} x (x + 3) + C

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What is the integral of (x+1)/(2sqrt(x)) ?
The integral of (x+1)/(2sqrt(x)) is 1/3 sqrt(x)(x+3)+C