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

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

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

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

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

Solution steps

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

Expand

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

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

2^{\frac{3}{2}} = 22

= \int x^{2} + 22 x^{\frac{3}{2}} + 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 22 x^{\frac{3}{2}} d x + \int 2 x d x

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

\int 22 x^{\frac{3}{2}} d x = \frac{4 2}{5} x^{\frac{5}{2}}

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

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

Add a constant to the solution

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

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