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

The integral of (x-1)/((2x+5)(x-8)) is 1/6 ln|2x+5|+1/3 ln|x-8|+C

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

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

\frac{1}{6} ln ∣ 2 x + 5 ∣ + \frac{1}{3} ln ∣ x - 8 ∣ + C

Solution steps

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

Expand

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

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

\int \frac{x}{( 2 x + 5 ) ( x - 8 )} d x = \frac{5}{42} ln ∣ 2 x + 5 ∣ + \frac{8}{21} ln ∣ x - 8 ∣

\int \frac{1}{( 2 x + 5 ) ( x - 8 )} d x = - \frac{1}{21} ln ∣ 2 x + 5 ∣ + \frac{1}{21} ln ∣ x - 8 ∣

= \frac{5}{42} ln ∣ 2 x + 5 ∣ + \frac{8}{21} ln ∣ x - 8 ∣ - (- \frac{1}{21} ln ∣ 2 x + 5 ∣ + \frac{1}{21} ln ∣ x - 8 ∣)

Simplify

\frac{5}{42} ln ∣ 2 x + 5 ∣ + \frac{8}{21} ln ∣ x - 8 ∣ - (- \frac{1}{21} ln ∣ 2 x + 5 ∣ + \frac{1}{21} ln ∣ x - 8 ∣) : \frac{1}{6} ln ∣ 2 x + 5 ∣ + \frac{1}{3} ln ∣ x - 8 ∣

= \frac{1}{6} ln ∣ 2 x + 5 ∣ + \frac{1}{3} ln ∣ x - 8 ∣

Add a constant to the solution

= \frac{1}{6} ln ∣ 2 x + 5 ∣ + \frac{1}{3} ln ∣ x - 8 ∣ + C

\int \frac{1}{t ^{2} 289 - t ^{2}} d t

\int_{- 7}^{7} \frac{1}{2} (49 - x^{2}) d x

t \to π lim (sin (\frac{t}{2}))

\frac{10}{s \cdot ( s + 1 ) \cdot ( s + 10 )}

What is the integral of (x-1)/((2x+5)(x-8)) ?
The integral of (x-1)/((2x+5)(x-8)) is 1/6 ln|2x+5|+1/3 ln|x-8|+C