What is the integral of (x^2-24)/(x(x+2)^3) ?

The integral of (x^2-24)/(x(x+2)^3) is -7/(x+2)-5/((x+2)^2)-3ln(1/2 |x|)+3ln| x/2+1|+C

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integral of (x^2-24)/(x(x+2)^3)

\int \frac{x ^{2} - 24}{x ( x + 2 ) ^{3}} d x

- \frac{7}{x + 2} - \frac{5}{( x + 2 ) ^{2}} - 3 ln (\frac{1}{2} ∣ x ∣) + 3 ln \frac{x}{2} + 1 + C

Solution steps

\int \frac{x ^{2} - 24}{x ( x + 2 ) ^{3}} d x

Expand

\frac{x ^{2} - 24}{x ( x + 2 ) ^{3}} : \frac{x}{( x + 2 ) ^{3}} - \frac{24}{x ( x + 2 ) ^{3}}

= \int \frac{x}{( x + 2 ) ^{3}} - \frac{24}{x ( x + 2 ) ^{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{x}{( x + 2 ) ^{3}} d x - \int \frac{24}{x ( x + 2 ) ^{3}} d x

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

\int \frac{24}{x ( x + 2 ) ^{3}} d x = 6 (\frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}} + \frac{1}{2} ln (\frac{1}{2} ∣ x ∣) - \frac{1}{2} ln \frac{x}{2} + 1)

= - \frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}} - 6 (\frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}} + \frac{1}{2} ln (\frac{1}{2} ∣ x ∣) - \frac{1}{2} ln \frac{x}{2} + 1)

Simplify

- \frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}} - 6 (\frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}} + \frac{1}{2} ln (\frac{1}{2} ∣ x ∣) - \frac{1}{2} ln \frac{x}{2} + 1) : - \frac{7}{x + 2} - \frac{5}{( x + 2 ) ^{2}} - 3 ln (\frac{1}{2} ∣ x ∣) + 3 ln \frac{x}{2} + 1

= - \frac{7}{x + 2} - \frac{5}{( x + 2 ) ^{2}} - 3 ln (\frac{1}{2} ∣ x ∣) + 3 ln \frac{x}{2} + 1

Add a constant to the solution

= - \frac{7}{x + 2} - \frac{5}{( x + 2 ) ^{2}} - 3 ln (\frac{1}{2} ∣ x ∣) + 3 ln \frac{x}{2} + 1 + C

What is the integral of (x^2-24)/(x(x+2)^3) ?
The integral of (x^2-24)/(x(x+2)^3) is -7/(x+2)-5/((x+2)^2)-3ln(1/2 |x|)+3ln| x/2+1|+C