What is the integral of ((x-95))/(x^2(x+5)) ?

The integral of ((x-95))/(x^2(x+5)) is 4ln|x|+(19)/x-4ln|x+5|+C

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integral of ((x-95))/(x^2(x+5))

\int \frac{( x - 95 )}{x ^{2} ( x + 5 )} d x

4 ln ∣ x ∣ + \frac{19}{x} - 4 ln ∣ x + 5 ∣ + C

Solution steps

\int \frac{x - 95}{x ^{2} ( x + 5 )} d x

Take the partial fraction of

\frac{x - 95}{x ^{2} ( x + 5 )} : \frac{4}{x} - \frac{19}{x ^{2}} - \frac{4}{x + 5}

= \int \frac{4}{x} - \frac{19}{x ^{2}} - \frac{4}{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{4}{x} d x - \int \frac{19}{x ^{2}} d x - \int \frac{4}{x + 5} d x

\int \frac{4}{x} d x = 4 ln ∣ x ∣

\int \frac{19}{x ^{2}} d x = - \frac{19}{x}

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

= 4 ln ∣ x ∣ - (- \frac{19}{x}) - 4 ln ∣ x + 5 ∣

Simplify

= 4 ln ∣ x ∣ + \frac{19}{x} - 4 ln ∣ x + 5 ∣

Add a constant to the solution

= 4 ln ∣ x ∣ + \frac{19}{x} - 4 ln ∣ x + 5 ∣ + C

y^{^{'}} = (x - 3) (y + 1)^{(\frac{2}{3})}

x \to 2 + lim (e^{\frac{3}{2 - x}})

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

\int 23

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

What is the integral of ((x-95))/(x^2(x+5)) ?
The integral of ((x-95))/(x^2(x+5)) is 4ln|x|+(19)/x-4ln|x+5|+C