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

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

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

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

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

Solution steps

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

Apply Integration By Parts

: (x^{2} - x + 2) \cdot 2 x^{\frac{1}{2}} - \int (2 x - 1) \cdot 2 x^{\frac{1}{2}} d x

= (x^{2} - x + 2) \cdot 2 x^{\frac{1}{2}} - \int (2 x - 1) \cdot 2 x^{\frac{1}{2}} d x

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

= (x^{2} - x + 2) \cdot 2 x^{\frac{1}{2}} - 2 (\frac{4}{5} x^{\frac{5}{2}} - \frac{2}{3} x^{\frac{3}{2}})

Simplify

(x^{2} - x + 2) \cdot 2 x^{\frac{1}{2}} - 2 (\frac{4}{5} x^{\frac{5}{2}} - \frac{2}{3} x^{\frac{3}{2}}) : \frac{2}{5} x^{\frac{5}{2}} - \frac{2}{3} x^{\frac{3}{2}} + 4 x^{\frac{1}{2}}

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

Add a constant to the solution

= \frac{2}{5} x^{\frac{5}{2}} - \frac{2}{3} x^{\frac{3}{2}} + 4 x^{\frac{1}{2}} + C

\frac{x}{( 1 + x ^{2} )} d x + \frac{y}{( 1 + y ^{2} )} d y = 0

f (x) = \frac{2 x}{1 + x ^{2}}

63 x^{2} y^{^{''}} + 14 x (x - 9) y^{^{'}} - 14 (x - 9) y = - 2 x^{3}

y = x^{2} - x

x \to 0 lim (csc (0))

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