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

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

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

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

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

Solution steps

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

Apply u-substitution

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

Apply the Power Rule

= \frac{1}{2} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = x + 2

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

Simplify

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

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

Add a constant to the solution

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

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