What is the integral of (1-1/(x^2))sqrt(x) ?

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

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integral of (1-1/(x^2))sqrt(x)

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

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

Solution steps

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

Expand

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

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

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

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

= - (- \frac{2}{x}) + \frac{2}{3} x^{\frac{3}{2}}

Simplify

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

Add a constant to the solution

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

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

e^{- 3 t} sin (2 t)

\frac{d}{d x} (\frac{x ^{2}}{9})

y^{^{''}} + 4 y = 2 csc^{2} (2 t)

\frac{d}{d y} (2 y^{2} e^{x y^{2}})

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