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

The integral of ((x+2)(x+3))/(x^2) is ((x+2)(x+3))/x+5ln|x|-(12)/x+C

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integral of ((x+2)(x+3))/(x^2)

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

\frac{( x + 2 ) ( x + 3 )}{x} + 5 ln ∣ x ∣ - \frac{12}{x} + C

Solution steps

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

Apply Integration By Parts

: \frac{( x + 2 ) ( x + 3 )}{x} - \int \frac{- 5 x - 12}{x ^{2}} d x

= \frac{( x + 2 ) ( x + 3 )}{x} - \int \frac{- 5 x - 12}{x ^{2}} d x

\int \frac{- 5 x - 12}{x ^{2}} d x = - 5 ln ∣ x ∣ + \frac{12}{x}

= \frac{( x + 2 ) ( x + 3 )}{x} - (- 5 ln ∣ x ∣ + \frac{12}{x})

Simplify

= \frac{( x + 2 ) ( x + 3 )}{x} + 5 ln ∣ x ∣ - \frac{12}{x}

Add a constant to the solution

= \frac{( x + 2 ) ( x + 3 )}{x} + 5 ln ∣ x ∣ - \frac{12}{x} + C

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What is the integral of ((x+2)(x+3))/(x^2) ?
The integral of ((x+2)(x+3))/(x^2) is ((x+2)(x+3))/x+5ln|x|-(12)/x+C