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

The integral of ((sqrt(x)+2)^3)/(2sqrt(x)) is 1/4 (sqrt(x)+2)^4+C

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

\int \frac{( x + 2 ) ^{3}}{2 x} d x

\frac{1}{4} (x + 2)^{4} + C

Solution steps

\int \frac{( x + 2 ) ^{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 + 2 ) ^{3}}{x} d x

Apply u-substitution

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

Apply the Power Rule

= \frac{1}{2} \cdot 2 \cdot \frac{u ^{4}}{4}

Substitute back

u = x + 2

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

Simplify

\frac{1}{2} \cdot 2 \cdot \frac{( x + 2 ) ^{4}}{4} : \frac{1}{4} (x + 2)^{4}

= \frac{1}{4} (x + 2)^{4}

Add a constant to the solution

= \frac{1}{4} (x + 2)^{4} + C

What is the integral of ((sqrt(x)+2)^3)/(2sqrt(x)) ?
The integral of ((sqrt(x)+2)^3)/(2sqrt(x)) is 1/4 (sqrt(x)+2)^4+C