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

The integral of (x^3+x)sqrt(x) is 2/9 x^{9/2}+2/5 x^{5/2}+C

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

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

\frac{2}{9} x^{\frac{9}{2}} + \frac{2}{5} x^{\frac{5}{2}} + C

Solution steps

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

Expand

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

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

\int x^{\frac{7}{2}} d x = \frac{2}{9} x^{\frac{9}{2}}

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

= \frac{2}{9} x^{\frac{9}{2}} + \frac{2}{5} x^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{9} x^{\frac{9}{2}} + \frac{2}{5} x^{\frac{5}{2}} + C

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What is the integral of (x^3+x)sqrt(x) ?
The integral of (x^3+x)sqrt(x) is 2/9 x^{9/2}+2/5 x^{5/2}+C