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

The integral of (2x^2-8)/(x(x+2)^3) is -4/(x+2)-ln(1/2 |x|)+ln| x/2+1|+C

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integral of (2x^2-8)/(x(x+2)^3)

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

- \frac{4}{x + 2} - ln (\frac{1}{2} ∣ x ∣) + ln \frac{x}{2} + 1 + C

Solution steps

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

Expand

\frac{2 x ^{2} - 8}{x ( x + 2 ) ^{3}} : \frac{2 x ^{2}}{x ( x + 2 ) ^{3}} - \frac{8}{x ( x + 2 ) ^{3}}

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

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

\int \frac{8}{x ( x + 2 ) ^{3}} d x = \frac{2}{x + 2} + \frac{2}{( x + 2 ) ^{2}} + ln (\frac{1}{2} ∣ x ∣) - ln \frac{x}{2} + 1

= 2 (- \frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}}) - (\frac{2}{x + 2} + \frac{2}{( x + 2 ) ^{2}} + ln (\frac{1}{2} ∣ x ∣) - ln \frac{x}{2} + 1)

Simplify

2 (- \frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}}) - (\frac{2}{x + 2} + \frac{2}{( x + 2 ) ^{2}} + ln (\frac{1}{2} ∣ x ∣) - ln \frac{x}{2} + 1) : - \frac{4}{x + 2} - ln (\frac{1}{2} ∣ x ∣) + ln \frac{x}{2} + 1

= - \frac{4}{x + 2} - ln (\frac{1}{2} ∣ x ∣) + ln \frac{x}{2} + 1

Add a constant to the solution

= - \frac{4}{x + 2} - ln (\frac{1}{2} ∣ x ∣) + ln \frac{x}{2} + 1 + C

12 x^{3} - 3 y^{2} x^{4} + 1 y^{^{'}} = 0

\int \frac{2}{5 x} d x

\int (sec (4 x) - cot (4 x)) d x

\frac{d}{d x} (sin (2 π x))

\frac{\partial}{\partial x} (x^{3} + y^{2})

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