What is the integral of sqrt(x^3+2x^2+x) ?

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

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integral of sqrt(x^3+2x^2+x)

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

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

Solution steps

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

Simplify

x^{3} + 2 x^{2} + x : x (x + 1)

= \int x (x + 1) d x

Expand

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

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

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

Add a constant to the solution

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

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