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

The integral of sqrt(x)*(3x+sqrt(x)) is 6/5 x^{5/2}+(x^2)/2+C

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

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

\frac{6}{5} x^{\frac{5}{2}} + \frac{x ^{2}}{2} + C

Solution steps

\int x (3 x + x) d x

Expand

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

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

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

\int 3 x^{\frac{3}{2}} d x = \frac{6}{5} x^{\frac{5}{2}}

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

= \frac{6}{5} x^{\frac{5}{2}} + \frac{x ^{2}}{2}

Add a constant to the solution

= \frac{6}{5} x^{\frac{5}{2}} + \frac{x ^{2}}{2} + C

What is the integral of sqrt(x)*(3x+sqrt(x)) ?
The integral of sqrt(x)*(3x+sqrt(x)) is 6/5 x^{5/2}+(x^2)/2+C