What is the integral of (x+3)/(12x^2-31x+20) ?

The integral of (x+3)/(12x^2-31x+20) is -5/4 ln|4x-5|+4/3 ln|3x-4|-3ln|24x-30|+3ln|24x-32|+5/4-4/3+C

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integral of (x+3)/(12x^2-31x+20)

\int \frac{x + 3}{12 x ^{2} - 31 x + 20} d x

- \frac{5}{4} ln ∣ 4 x - 5 ∣ + \frac{4}{3} ln ∣ 3 x - 4 ∣ - 3 ln ∣ 24 x - 30 ∣ + 3 ln ∣ 24 x - 32 ∣ + \frac{5}{4} - \frac{4}{3} + C

Solution steps

\int \frac{x + 3}{12 x ^{2} - 31 x + 20} d x

Apply Integration By Parts

: (x + 3) (- ln ∣ 24 x - 30 ∣ + ln ∣ 24 x - 32 ∣) - \int - ln ∣ 24 x - 30 ∣ + ln ∣ 24 x - 32 ∣ d x

= (x + 3) (- ln ∣ 24 x - 30 ∣ + ln ∣ 24 x - 32 ∣) - \int - ln ∣ 24 x - 30 ∣ + ln ∣ 24 x - 32 ∣ d x

\int - ln ∣ 24 x - 30 ∣ + ln ∣ 24 x - 32 ∣ d x = - x ln ∣ 24 x - 30 ∣ + x ln ∣ 24 x - 32 ∣ + \frac{5}{4} ln ∣ 4 x - 5 ∣ - \frac{4}{3} ln ∣ 3 x - 4 ∣ - \frac{5}{4} + \frac{4}{3}

= (x + 3) (- ln ∣ 24 x - 30 ∣ + ln ∣ 24 x - 32 ∣) - (- x ln ∣ 24 x - 30 ∣ + x ln ∣ 24 x - 32 ∣ + \frac{5}{4} ln ∣ 4 x - 5 ∣ - \frac{4}{3} ln ∣ 3 x - 4 ∣ - \frac{5}{4} + \frac{4}{3})

Simplify

(x + 3) (- ln ∣ 24 x - 30 ∣ + ln ∣ 24 x - 32 ∣) - (- x ln ∣ 24 x - 30 ∣ + x ln ∣ 24 x - 32 ∣ + \frac{5}{4} ln ∣ 4 x - 5 ∣ - \frac{4}{3} ln ∣ 3 x - 4 ∣ - \frac{5}{4} + \frac{4}{3}) : - \frac{5}{4} ln ∣ 4 x - 5 ∣ + \frac{4}{3} ln ∣ 3 x - 4 ∣ - 3 ln ∣ 24 x - 30 ∣ + 3 ln ∣ 24 x - 32 ∣ + \frac{5}{4} - \frac{4}{3}

= - \frac{5}{4} ln ∣ 4 x - 5 ∣ + \frac{4}{3} ln ∣ 3 x - 4 ∣ - 3 ln ∣ 24 x - 30 ∣ + 3 ln ∣ 24 x - 32 ∣ + \frac{5}{4} - \frac{4}{3}

Add a constant to the solution

= - \frac{5}{4} ln ∣ 4 x - 5 ∣ + \frac{4}{3} ln ∣ 3 x - 4 ∣ - 3 ln ∣ 24 x - 30 ∣ + 3 ln ∣ 24 x - 32 ∣ + \frac{5}{4} - \frac{4}{3} + C

\int \frac{x ^{4} - 6 x ^{3} - 7 x}{x} d x

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

\int x (x - 1) d x

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

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

What is the integral of (x+3)/(12x^2-31x+20) ?
The integral of (x+3)/(12x^2-31x+20) is -5/4 ln|4x-5|+4/3 ln|3x-4|-3ln|24x-30|+3ln|24x-32|+5/4-4/3+C