What is the integral of (x+9)/(5x^2-8x+3) ?

The integral of (x+9)/(5x^2-8x+3) is -24/5 ln|5x-3|+5ln|5x-5|-1/5+C

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integral of (x+9)/(5x^2-8x+3)

\int \frac{x + 9}{5 x ^{2} - 8 x + 3} d x

- \frac{24}{5} ln ∣ 5 x - 3 ∣ + 5 ln ∣ 5 x - 5 ∣ - \frac{1}{5} + C

Solution steps

\int \frac{x + 9}{5 x ^{2} - 8 x + 3} d x

Apply Integration By Parts

: (x + 9) (- \frac{1}{2} ln ∣ 5 x - 3 ∣ + \frac{1}{2} ln ∣ 5 x - 5 ∣) - \int \frac{1}{2} (- ln ∣ 5 x - 3 ∣ + ln ∣ 5 x - 5 ∣) d x

= (x + 9) (- \frac{1}{2} ln ∣ 5 x - 3 ∣ + \frac{1}{2} ln ∣ 5 x - 5 ∣) - \int \frac{1}{2} (- ln ∣ 5 x - 3 ∣ + ln ∣ 5 x - 5 ∣) d x

\int \frac{1}{2} (- ln ∣ 5 x - 3 ∣ + ln ∣ 5 x - 5 ∣) d x = \frac{1}{10} (5 x ln ∣ 5 x - 5 ∣ - 5 x ln ∣ 5 x - 3 ∣ + 3 ln ∣ 5 x - 3 ∣ - 5 ln ∣ 5 x - 5 ∣ + 2)

= (x + 9) (- \frac{1}{2} ln ∣ 5 x - 3 ∣ + \frac{1}{2} ln ∣ 5 x - 5 ∣) - \frac{1}{10} (5 x ln ∣ 5 x - 5 ∣ - 5 x ln ∣ 5 x - 3 ∣ + 3 ln ∣ 5 x - 3 ∣ - 5 ln ∣ 5 x - 5 ∣ + 2)

Simplify

(x + 9) (- \frac{1}{2} ln ∣ 5 x - 3 ∣ + \frac{1}{2} ln ∣ 5 x - 5 ∣) - \frac{1}{10} (5 x ln ∣ 5 x - 5 ∣ - 5 x ln ∣ 5 x - 3 ∣ + 3 ln ∣ 5 x - 3 ∣ - 5 ln ∣ 5 x - 5 ∣ + 2) : - \frac{24}{5} ln ∣ 5 x - 3 ∣ + 5 ln ∣ 5 x - 5 ∣ - \frac{1}{5}

= - \frac{24}{5} ln ∣ 5 x - 3 ∣ + 5 ln ∣ 5 x - 5 ∣ - \frac{1}{5}

Add a constant to the solution

= - \frac{24}{5} ln ∣ 5 x - 3 ∣ + 5 ln ∣ 5 x - 5 ∣ - \frac{1}{5} + C

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\int \frac{1}{9 s - 8} d s

What is the integral of (x+9)/(5x^2-8x+3) ?
The integral of (x+9)/(5x^2-8x+3) is -24/5 ln|5x-3|+5ln|5x-5|-1/5+C