What is the integral of (5x+4)/((x+8)^2(x-1)) ?

The integral of (5x+4)/((x+8)^2(x-1)) is -1/9 ln|x+8|+1/9 ln|x-1|-4/(x+8)+C

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integral of (5x+4)/((x+8)^2(x-1))

\int \frac{5 x + 4}{( x + 8 ) ^{2} ( x - 1 )} d x

- \frac{1}{9} ln ∣ x + 8 ∣ + \frac{1}{9} ln ∣ x - 1 ∣ - \frac{4}{x + 8} + C

Solution steps

\int \frac{5 x + 4}{( x + 8 ) ^{2} ( x - 1 )} d x

Expand

\frac{5 x + 4}{( x + 8 ) ^{2} ( x - 1 )} : \frac{5 x}{( x + 8 ) ^{2} ( x - 1 )} + \frac{4}{( x + 8 ) ^{2} ( x - 1 )}

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 x}{( x + 8 ) ^{2} ( x - 1 )} d x + \int \frac{4}{( x + 8 ) ^{2} ( x - 1 )} d x

\int \frac{5 x}{( x + 8 ) ^{2} ( x - 1 )} d x = 5 (- \frac{1}{81} ln ∣ x + 8 ∣ - \frac{8}{9 ( x + 8 )} + \frac{1}{81} ln ∣ x - 1 ∣)

\int \frac{4}{( x + 8 ) ^{2} ( x - 1 )} d x = 4 (- \frac{1}{81} ln ∣ x + 8 ∣ + \frac{1}{9 ( x + 8 )} + \frac{1}{81} ln ∣ x - 1 ∣)

= 5 (- \frac{1}{81} ln ∣ x + 8 ∣ - \frac{8}{9 ( x + 8 )} + \frac{1}{81} ln ∣ x - 1 ∣) + 4 (- \frac{1}{81} ln ∣ x + 8 ∣ + \frac{1}{9 ( x + 8 )} + \frac{1}{81} ln ∣ x - 1 ∣)

Simplify

5 (- \frac{1}{81} ln ∣ x + 8 ∣ - \frac{8}{9 ( x + 8 )} + \frac{1}{81} ln ∣ x - 1 ∣) + 4 (- \frac{1}{81} ln ∣ x + 8 ∣ + \frac{1}{9 ( x + 8 )} + \frac{1}{81} ln ∣ x - 1 ∣) : - \frac{1}{9} ln ∣ x + 8 ∣ + \frac{1}{9} ln ∣ x - 1 ∣ - \frac{4}{x + 8}

= - \frac{1}{9} ln ∣ x + 8 ∣ + \frac{1}{9} ln ∣ x - 1 ∣ - \frac{4}{x + 8}

Add a constant to the solution

= - \frac{1}{9} ln ∣ x + 8 ∣ + \frac{1}{9} ln ∣ x - 1 ∣ - \frac{4}{x + 8} + C

\frac{s ^{2} + 9}{s ( s + 4 )}

\frac{d y}{d x} = \frac{x - 1}{x + 3} \frac{y - 2}{y - 3}

\frac{2 s + 3}{s ^{2} + 6 s + 13}

\frac{x}{x - 2}

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

What is the integral of (5x+4)/((x+8)^2(x-1)) ?
The integral of (5x+4)/((x+8)^2(x-1)) is -1/9 ln|x+8|+1/9 ln|x-1|-4/(x+8)+C