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

The integral of (x+sqrt(x))^2 is (x^3)/3+4/5 x^{5/2}+(x^2)/2+C

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

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

\frac{x ^{3}}{3} + \frac{4}{5} x^{\frac{5}{2}} + \frac{x ^{2}}{2} + C

Solution steps

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

Expand

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

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

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

\int 2 x^{\frac{3}{2}} d x = \frac{4}{5} x^{\frac{5}{2}}

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

= \frac{x ^{3}}{3} + \frac{4}{5} x^{\frac{5}{2}} + \frac{x ^{2}}{2}

Add a constant to the solution

= \frac{x ^{3}}{3} + \frac{4}{5} x^{\frac{5}{2}} + \frac{x ^{2}}{2} + C

What is the integral of (x+sqrt(x))^2 ?
The integral of (x+sqrt(x))^2 is (x^3)/3+4/5 x^{5/2}+(x^2)/2+C