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

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

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integral of (sqrt(x)+3)/(2sqrt(x))

\int \frac{x + 3}{2 x} d x

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

Solution steps

\int \frac{x + 3}{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 + 3}{x} d x

Apply u-substitution

: \int 2 u d u

= \frac{1}{2} \cdot \int 2 u d u

Take the constant out:

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

= \frac{1}{2} \cdot 2 \cdot \int u d u

Apply the Power Rule

: \frac{u ^{2}}{2}

= \frac{1}{2} \cdot 2 \cdot \frac{u ^{2}}{2}

Substitute back

u = x + 3

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

Simplify

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

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

Add a constant to the solution

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

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