What is the integral of (x^4+x)/(sqrt(x)) ?

The integral of (x^4+x)/(sqrt(x)) is 2/9 x^{9/2}+2/3 x^{3/2}+C

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integral of (x^4+x)/(sqrt(x))

\int \frac{x ^{4} + x}{x} d x

\frac{2}{9} x^{\frac{9}{2}} + \frac{2}{3} x^{\frac{3}{2}} + C

Solution steps

\int \frac{x ^{4} + x}{x} d x

Simplify

\frac{x ^{4} + x}{x} : x (x^{3} + 1)

= \int x (x^{3} + 1) d x

Expand

x (x^{3} + 1) : x^{\frac{7}{2}} + x

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

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

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

= \frac{2}{9} x^{\frac{9}{2}} + \frac{2}{3} x^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{9} x^{\frac{9}{2}} + \frac{2}{3} x^{\frac{3}{2}} + C

What is the integral of (x^4+x)/(sqrt(x)) ?
The integral of (x^4+x)/(sqrt(x)) is 2/9 x^{9/2}+2/3 x^{3/2}+C