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

The integral of sqrt(x)(x^2+3x+2) is 2/7 x^{7/2}+6/5 x^{5/2}+4/3 x^{3/2}+C

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

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

\frac{2}{7} x^{\frac{7}{2}} + \frac{6}{5} x^{\frac{5}{2}} + \frac{4}{3} x^{\frac{3}{2}} + C

Solution steps

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

Expand

x (x^{2} + 3 x + 2) : x^{\frac{5}{2}} + 3 x^{\frac{3}{2}} + 2 x

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

\int x^{\frac{5}{2}} d x = \frac{2}{7} x^{\frac{7}{2}}

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

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

= \frac{2}{7} x^{\frac{7}{2}} + \frac{6}{5} x^{\frac{5}{2}} + \frac{4}{3} x^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{7} x^{\frac{7}{2}} + \frac{6}{5} x^{\frac{5}{2}} + \frac{4}{3} x^{\frac{3}{2}} + C

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What is the integral of sqrt(x)(x^2+3x+2) ?
The integral of sqrt(x)(x^2+3x+2) is 2/7 x^{7/2}+6/5 x^{5/2}+4/3 x^{3/2}+C