What is the integral of (8x+6)/((4x^2+6x)^3) ?

The integral of (8x+6)/((4x^2+6x)^3) is 1/(54x)-1/(72x^2)-1/(27(2x+3))-1/(18(2x+3)^2)+C

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integral of (8x+6)/((4x^2+6x)^3)

\int \frac{8 x + 6}{( 4 x ^{2} + 6 x ) ^{3}} d x

\frac{1}{54 x} - \frac{1}{72 x ^{2}} - \frac{1}{27 ( 2 x + 3 )} - \frac{1}{18 ( 2 x + 3 ) ^{2}} + C

Solution steps

\int \frac{8 x + 6}{( 4 x ^{2} + 6 x ) ^{3}} d x

Take the partial fraction of

\frac{8 x + 6}{( 4 x ^{2} + 6 x ) ^{3}} : - \frac{1}{54 x ^{2}} + \frac{1}{36 x ^{3}} + \frac{2}{27 ( 2 x + 3 ) ^{2}} + \frac{2}{9 ( 2 x + 3 ) ^{3}}

= \int - \frac{1}{54 x ^{2}} + \frac{1}{36 x ^{3}} + \frac{2}{27 ( 2 x + 3 ) ^{2}} + \frac{2}{9 ( 2 x + 3 ) ^{3}} 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{1}{54 x ^{2}} d x + \int \frac{1}{36 x ^{3}} d x + \int \frac{2}{27 ( 2 x + 3 ) ^{2}} d x + \int \frac{2}{9 ( 2 x + 3 ) ^{3}} d x

\int \frac{1}{54 x ^{2}} d x = - \frac{1}{54 x}

\int \frac{1}{36 x ^{3}} d x = - \frac{1}{72 x ^{2}}

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

\int \frac{2}{9 ( 2 x + 3 ) ^{3}} d x = - \frac{1}{18 ( 2 x + 3 ) ^{2}}

= - (- \frac{1}{54 x}) - \frac{1}{72 x ^{2}} - \frac{1}{27 ( 2 x + 3 )} - \frac{1}{18 ( 2 x + 3 ) ^{2}}

Simplify

= \frac{1}{54 x} - \frac{1}{72 x ^{2}} - \frac{1}{27 ( 2 x + 3 )} - \frac{1}{18 ( 2 x + 3 ) ^{2}}

Add a constant to the solution

= \frac{1}{54 x} - \frac{1}{72 x ^{2}} - \frac{1}{27 ( 2 x + 3 )} - \frac{1}{18 ( 2 x + 3 ) ^{2}} + C

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What is the integral of (8x+6)/((4x^2+6x)^3) ?
The integral of (8x+6)/((4x^2+6x)^3) is 1/(54x)-1/(72x^2)-1/(27(2x+3))-1/(18(2x+3)^2)+C