What is the integral of (x^3+x^{-3})/(sqrt(x)) ?

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

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integral of (x^3+x^{-3})/(sqrt(x))

\int \frac{x ^{3} + x ^{- 3}}{x} d x

\frac{2}{7} x^{\frac{7}{2}} - \frac{2}{5 x ^{\frac{5}{2}}} + C

Solution steps

\int \frac{x ^{3} + x ^{- 3}}{x} d x

Expand

\frac{x ^{3} + x ^{- 3}}{x} : x^{\frac{5}{2}} + \frac{1}{x ^{3} x}

= \int x^{\frac{5}{2}} + \frac{1}{x ^{3} 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 \frac{1}{x ^{3} x} d x

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

\int \frac{1}{x ^{3} x} d x = - \frac{2}{5 x ^{\frac{5}{2}}}

= \frac{2}{7} x^{\frac{7}{2}} - \frac{2}{5 x ^{\frac{5}{2}}}

Add a constant to the solution

= \frac{2}{7} x^{\frac{7}{2}} - \frac{2}{5 x ^{\frac{5}{2}}} + C

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