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

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

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

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

\frac{2}{3} x^{\frac{3}{2}} + \frac{2}{x ^{\frac{1}{2}}} + C

Solution steps

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

Expand

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

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

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

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

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

Simplify

= \frac{2}{3} x^{\frac{3}{2}} + \frac{2}{x ^{\frac{1}{2}}}

Add a constant to the solution

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

\int_{0}^{\frac{π}{6}} sin^{2} (x) d x

x \to 0 lim (\frac{5 ^{x} - 1 0 ^{x}}{x})

x \to 5 lim (x^{3} + 2 x)

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

y^{\frac{1}{2}} \frac{d y}{d x} + y^{\frac{3}{2}} = 1, y (0) = 9

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