What is the integral of sqrt(x)(x-sqrt(x)+1) ?

The integral of sqrt(x)(x-sqrt(x)+1) is 2/5 x^{5/2}-(x^2)/2+2/3 x^{3/2}+C

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integral of sqrt(x)(x-sqrt(x)+1)

\int x (x - x + 1) d x

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

Solution steps

\int x (x - x + 1) d x

Expand

x (x - x + 1) : x^{\frac{3}{2}} - x + x

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

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

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

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

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

Add a constant to the solution

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

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What is the integral of sqrt(x)(x-sqrt(x)+1) ?
The integral of sqrt(x)(x-sqrt(x)+1) is 2/5 x^{5/2}-(x^2)/2+2/3 x^{3/2}+C