What is the integral of (3x^5+4x^{3/2}-2x^{-1/2}) ?

The integral of (3x^5+4x^{3/2}-2x^{-1/2}) is (x^6)/2+8/5 x^{5/2}-4x^{1/2}+C

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integral of (3x^5+4x^{3/2}-2x^{-1/2})

\int (3 x^{5} + 4 x^{\frac{3}{2}} - 2 x^{- \frac{1}{2}}) d x

\frac{x ^{6}}{2} + \frac{8}{5} x^{\frac{5}{2}} - 4 x^{\frac{1}{2}} + C

Solution steps

\int 3 x^{5} + 4 x^{\frac{3}{2}} - 2 x^{- \frac{1}{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 3 x^{5} d x + \int 4 x^{\frac{3}{2}} d x - \int 2 x^{- \frac{1}{2}} d x

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

\int 4 x^{\frac{3}{2}} d x = \frac{8}{5} x^{\frac{5}{2}}

\int 2 x^{- \frac{1}{2}} d x = 4 x^{\frac{1}{2}}

= \frac{x ^{6}}{2} + \frac{8}{5} x^{\frac{5}{2}} - 4 x^{\frac{1}{2}}

Add a constant to the solution

= \frac{x ^{6}}{2} + \frac{8}{5} x^{\frac{5}{2}} - 4 x^{\frac{1}{2}} + C

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\frac{\partial}{\partial x} (\frac{- 8 x}{x ^{2} + 1})

What is the integral of (3x^5+4x^{3/2}-2x^{-1/2}) ?
The integral of (3x^5+4x^{3/2}-2x^{-1/2}) is (x^6)/2+8/5 x^{5/2}-4x^{1/2}+C