What is the integral of (6x+13)/(x^2+5x+6) ?

The integral of (6x+13)/(x^2+5x+6) is ln|x+2|+5ln|x+3|+C

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integral of (6x+13)/(x^2+5x+6)

\int \frac{6 x + 13}{x ^{2} + 5 x + 6} d x

ln ∣ x + 2 ∣ + 5 ln ∣ x + 3 ∣ + C

Solution steps

\int \frac{6 x + 13}{x ^{2} + 5 x + 6} d x

Take the partial fraction of

\frac{6 x + 13}{x ^{2} + 5 x + 6} : \frac{1}{x + 2} + \frac{5}{x + 3}

= \int \frac{1}{x + 2} + \frac{5}{x + 3} 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{1}{x + 2} d x + \int \frac{5}{x + 3} d x

\int \frac{1}{x + 2} d x = ln ∣ x + 2 ∣

\int \frac{5}{x + 3} d x = 5 ln ∣ x + 3 ∣

= ln ∣ x + 2 ∣ + 5 ln ∣ x + 3 ∣

Add a constant to the solution

= ln ∣ x + 2 ∣ + 5 ln ∣ x + 3 ∣ + C

\int cos^{3} (x) d x

\frac{ln ( x )}{x}

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

\int \frac{x ^{2}}{( x - 3 ) ^{5}} d x

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

What is the integral of (6x+13)/(x^2+5x+6) ?
The integral of (6x+13)/(x^2+5x+6) is ln|x+2|+5ln|x+3|+C