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

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

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

\int 3 x (x + 3) d x

\frac{3 x ^{2}}{2} + 2 x^{\frac{3}{2}} + C

Solution steps

\int 3 x (x + 3) d x

3 x = 3 x

= \int 3 x (x + 3) d x

Take the constant out:

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

= 3 \cdot \int x (x + 3) d x

Expand

x (x + 3) : x + 3 x

= 3 \cdot \int x + 3 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

= 3 (\int x d x + \int 3 x d x)

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

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

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

Simplify

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

= \frac{3 x ^{2}}{2} + 2 x^{\frac{3}{2}}

Add a constant to the solution

= \frac{3 x ^{2}}{2} + 2 x^{\frac{3}{2}} + C

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