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

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

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

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

\frac{5}{4} ln ∣ x ∣ + \frac{3}{2 x} - \frac{5}{4} ln ∣ x + 2 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{x - 3}{x ^{3} + 2 x ^{2}} : \frac{5}{4 x} - \frac{3}{2 x ^{2}} - \frac{5}{4 ( x + 2 )}

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

\int \frac{5}{4 x} d x = \frac{5}{4} ln ∣ x ∣

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

\int \frac{5}{4 ( x + 2 )} d x = \frac{5}{4} ln ∣ x + 2 ∣

= \frac{5}{4} ln ∣ x ∣ - (- \frac{3}{2 x}) - \frac{5}{4} ln ∣ x + 2 ∣

Simplify

= \frac{5}{4} ln ∣ x ∣ + \frac{3}{2 x} - \frac{5}{4} ln ∣ x + 2 ∣

Add a constant to the solution

= \frac{5}{4} ln ∣ x ∣ + \frac{3}{2 x} - \frac{5}{4} ln ∣ x + 2 ∣ + C

\frac{d}{d x} (5 x^{2} - 3 x + 2)

\frac{2}{x} - \frac{2}{x ^{3}} + \frac{1}{x ^{4}}

\int \frac{sin ( 2 x )}{1 - cos ( 2 x )} d x

\int (x^{2} e^{a x} + \frac{1}{a x}) d x

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