What is the integral of sqrt(u)(u^2+3u+7) ?

The integral of sqrt(u)(u^2+3u+7) is 2/7 u^{7/2}+6/5 u^{5/2}+14/3 u^{3/2}+C

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integral of sqrt(u)(u^2+3u+7)

\int u (u^{2} + 3 u + 7) d u

\frac{2}{7} u^{\frac{7}{2}} + \frac{6}{5} u^{\frac{5}{2}} + \frac{14}{3} u^{\frac{3}{2}} + C

Solution steps

\int u (u^{2} + 3 u + 7) d u

Expand

u (u^{2} + 3 u + 7) : u^{\frac{5}{2}} + 3 u^{\frac{3}{2}} + 7 u

= \int u^{\frac{5}{2}} + 3 u^{\frac{3}{2}} + 7 u d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u^{\frac{5}{2}} d u + \int 3 u^{\frac{3}{2}} d u + \int 7 u d u

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

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

\int 7 u d u = \frac{14}{3} u^{\frac{3}{2}}

= \frac{2}{7} u^{\frac{7}{2}} + \frac{6}{5} u^{\frac{5}{2}} + \frac{14}{3} u^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{7} u^{\frac{7}{2}} + \frac{6}{5} u^{\frac{5}{2}} + \frac{14}{3} u^{\frac{3}{2}} + C

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What is the integral of sqrt(u)(u^2+3u+7) ?
The integral of sqrt(u)(u^2+3u+7) is 2/7 u^{7/2}+6/5 u^{5/2}+14/3 u^{3/2}+C