What is the integral of (9x^2+x)/((12x^3+2x^2)^2) ?

The integral of (9x^2+x)/((12x^3+2x^2)^2) is 3/(4x)-1/(8x^2)-9/(2(6x+1))+C

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integral of (9x^2+x)/((12x^3+2x^2)^2)

\int \frac{9 x ^{2} + x}{( 12 x ^{3} + 2 x ^{2} ) ^{2}} d x

\frac{3}{4 x} - \frac{1}{8 x ^{2}} - \frac{9}{2 ( 6 x + 1 )} + C

Solution steps

\int \frac{9 x ^{2} + x}{( 12 x ^{3} + 2 x ^{2} ) ^{2}} d x

Take the partial fraction of

\frac{9 x ^{2} + x}{( 12 x ^{3} + 2 x ^{2} ) ^{2}} : - \frac{3}{4 x ^{2}} + \frac{1}{4 x ^{3}} + \frac{27}{( 6 x + 1 ) ^{2}}

= \int - \frac{3}{4 x ^{2}} + \frac{1}{4 x ^{3}} + \frac{27}{( 6 x + 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{3}{4 x ^{2}} d x + \int \frac{1}{4 x ^{3}} d x + \int \frac{27}{( 6 x + 1 ) ^{2}} d x

\int \frac{3}{4 x ^{2}} d x = - \frac{3}{4 x}

\int \frac{1}{4 x ^{3}} d x = - \frac{1}{8 x ^{2}}

\int \frac{27}{( 6 x + 1 ) ^{2}} d x = - \frac{9}{2 ( 6 x + 1 )}

= - (- \frac{3}{4 x}) - \frac{1}{8 x ^{2}} - \frac{9}{2 ( 6 x + 1 )}

Simplify

= \frac{3}{4 x} - \frac{1}{8 x ^{2}} - \frac{9}{2 ( 6 x + 1 )}

Add a constant to the solution

= \frac{3}{4 x} - \frac{1}{8 x ^{2}} - \frac{9}{2 ( 6 x + 1 )} + C

What is the integral of (9x^2+x)/((12x^3+2x^2)^2) ?
The integral of (9x^2+x)/((12x^3+2x^2)^2) is 3/(4x)-1/(8x^2)-9/(2(6x+1))+C