What is the integral of (2x-11)/(x^2(x+2)) ?

The integral of (2x-11)/(x^2(x+2)) is 15/4 ln|x|+(11)/(2x)-15/4 ln|x+2|+C

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integral of (2x-11)/(x^2(x+2))

\int \frac{2 x - 11}{x ^{2} ( x + 2 )} d x

\frac{15}{4} ln ∣ x ∣ + \frac{11}{2 x} - \frac{15}{4} ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{2 x - 11}{x ^{2} ( x + 2 )} d x

Take the partial fraction of

\frac{2 x - 11}{x ^{2} ( x + 2 )} : \frac{15}{4 x} - \frac{11}{2 x ^{2}} - \frac{15}{4 ( x + 2 )}

= \int \frac{15}{4 x} - \frac{11}{2 x ^{2}} - \frac{15}{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{15}{4 x} d x - \int \frac{11}{2 x ^{2}} d x - \int \frac{15}{4 ( x + 2 )} d x

\int \frac{15}{4 x} d x = \frac{15}{4} ln ∣ x ∣

\int \frac{11}{2 x ^{2}} d x = - \frac{11}{2 x}

\int \frac{15}{4 ( x + 2 )} d x = \frac{15}{4} ln ∣ x + 2 ∣

= \frac{15}{4} ln ∣ x ∣ - (- \frac{11}{2 x}) - \frac{15}{4} ln ∣ x + 2 ∣

Simplify

= \frac{15}{4} ln ∣ x ∣ + \frac{11}{2 x} - \frac{15}{4} ln ∣ x + 2 ∣

Add a constant to the solution

= \frac{15}{4} ln ∣ x ∣ + \frac{11}{2 x} - \frac{15}{4} ln ∣ x + 2 ∣ + C

y^{^{'}} - 2 y = 3 e^{x}

\int_{2}^{10} \frac{x ^{2} + 36}{12 x} d x

x^{^{'''}} - x^{^{''}} - x^{^{'}} + x = 2 e^{- t} + 3

\frac{\partial}{\partial x} (8 x - 2 y - 2)

\frac{\partial}{\partial y} (\frac{y}{z})

What is the integral of (2x-11)/(x^2(x+2)) ?
The integral of (2x-11)/(x^2(x+2)) is 15/4 ln|x|+(11)/(2x)-15/4 ln|x+2|+C