What is the integral of ((x-2)(x+3))/(xsqrt(x)) ?

The integral of ((x-2)(x+3))/(xsqrt(x)) is 2/3 x^{3/2}+2sqrt(x)+(12)/(sqrt(x))+C

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integral of ((x-2)(x+3))/(xsqrt(x))

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

\frac{2}{3} x^{\frac{3}{2}} + 2 x + \frac{12}{x} + C

Solution steps

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

Expand

\frac{( x - 2 ) ( x + 3 )}{x x} : x + \frac{1}{x} - \frac{6}{x x}

= \int x + \frac{1}{x} - \frac{6}{x 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 d x + \int \frac{1}{x} d x - \int \frac{6}{x x} d x

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

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

\int \frac{6}{x x} d x = - \frac{12}{x}

= \frac{2}{3} x^{\frac{3}{2}} + 2 x - (- \frac{12}{x})

Simplify

= \frac{2}{3} x^{\frac{3}{2}} + 2 x + \frac{12}{x}

Add a constant to the solution

= \frac{2}{3} x^{\frac{3}{2}} + 2 x + \frac{12}{x} + C

\int ln (x^{7}) d x

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

\int (2 a x + b) (a x^{2} + b x)^{7} d x

\int e^{- 3 x} sin (2 x) d x

x^{2} 5^{x}

What is the integral of ((x-2)(x+3))/(xsqrt(x)) ?
The integral of ((x-2)(x+3))/(xsqrt(x)) is 2/3 x^{3/2}+2sqrt(x)+(12)/(sqrt(x))+C