What is the integral of (2-x)/(x^2+5x) ?

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

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integral of (2-x)/(x^2+5x)

\int \frac{2 - x}{x ^{2} + 5 x} d x

\frac{2}{5} ln ∣ x ∣ - \frac{7}{5} ln ∣ x + 5 ∣ + C

Solution steps

\int \frac{2 - x}{x ^{2} + 5 x} d x

Take the partial fraction of

\frac{2 - x}{x ^{2} + 5 x} : \frac{2}{5 x} - \frac{7}{5 ( x + 5 )}

= \int \frac{2}{5 x} - \frac{7}{5 ( x + 5 )} 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{2}{5 x} d x - \int \frac{7}{5 ( x + 5 )} d x

\int \frac{2}{5 x} d x = \frac{2}{5} ln ∣ x ∣

\int \frac{7}{5 ( x + 5 )} d x = \frac{7}{5} ln ∣ x + 5 ∣

= \frac{2}{5} ln ∣ x ∣ - \frac{7}{5} ln ∣ x + 5 ∣

Add a constant to the solution

= \frac{2}{5} ln ∣ x ∣ - \frac{7}{5} ln ∣ x + 5 ∣ + C

x \to 0 lim (- \frac{9}{1 - 9 x})

\frac{\partial}{\partial y} (\frac{x ^{2}}{y ^{2} + 1})

\frac{\partial}{\partial y} (5 x^{2} y)

\frac{d}{d x} (\frac{x ^{2} - 4 x + 20}{x ^{2} + 6 x + 8})

f (x) = \frac{2 x ^{2} + 4 x + 4}{x}

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