What is the integral of ((2x+5))/(9x^2+12x+4) ?

The integral of ((2x+5))/(9x^2+12x+4) is 2(-x/(3(3x+2))+1/9 ln|3x+2|)-5/(3(3x+2))+C

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integral of ((2x+5))/(9x^2+12x+4)

\int \frac{( 2 x + 5 )}{9 x ^{2} + 12 x + 4} d x

2 (- \frac{x}{3 ( 3 x + 2 )} + \frac{1}{9} ln ∣ 3 x + 2 ∣) - \frac{5}{3 ( 3 x + 2 )} + C

Solution steps

\int \frac{2 x + 5}{9 x ^{2} + 12 x + 4} d x

Expand

\frac{2 x + 5}{9 x ^{2} + 12 x + 4} : \frac{2 x}{9 x ^{2} + 12 x + 4} + \frac{5}{9 x ^{2} + 12 x + 4}

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{2 x}{9 x ^{2} + 12 x + 4} d x + \int \frac{5}{9 x ^{2} + 12 x + 4} d x

\int \frac{2 x}{9 x ^{2} + 12 x + 4} d x = 2 (- \frac{x}{3 ( 3 x + 2 )} + \frac{1}{9} ln ∣ 3 x + 2 ∣)

\int \frac{5}{9 x ^{2} + 12 x + 4} d x = - \frac{5}{3 ( 3 x + 2 )}

= 2 (- \frac{x}{3 ( 3 x + 2 )} + \frac{1}{9} ln ∣ 3 x + 2 ∣) - \frac{5}{3 ( 3 x + 2 )}

Add a constant to the solution

= 2 (- \frac{x}{3 ( 3 x + 2 )} + \frac{1}{9} ln ∣ 3 x + 2 ∣) - \frac{5}{3 ( 3 x + 2 )} + C

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What is the integral of ((2x+5))/(9x^2+12x+4) ?
The integral of ((2x+5))/(9x^2+12x+4) is 2(-x/(3(3x+2))+1/9 ln|3x+2|)-5/(3(3x+2))+C