What is the integral of x^3(8x+sqrt(x)) ?

The integral of x^3(8x+sqrt(x)) is (8x^5)/5+2/9 x^{9/2}+C

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integral of x^3(8x+sqrt(x))

\int x^{3} (8 x + x) d x

\frac{8 x ^{5}}{5} + \frac{2}{9} x^{\frac{9}{2}} + C

Solution steps

\int x^{3} (8 x + x) d x

Expand

x^{3} (8 x + x) : 8 x^{4} + x^{\frac{7}{2}}

= \int 8 x^{4} + x^{\frac{7}{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 8 x^{4} d x + \int x^{\frac{7}{2}} d x

\int 8 x^{4} d x = \frac{8 x ^{5}}{5}

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

= \frac{8 x ^{5}}{5} + \frac{2}{9} x^{\frac{9}{2}}

Add a constant to the solution

= \frac{8 x ^{5}}{5} + \frac{2}{9} x^{\frac{9}{2}} + C

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What is the integral of x^3(8x+sqrt(x)) ?
The integral of x^3(8x+sqrt(x)) is (8x^5)/5+2/9 x^{9/2}+C