What is the integral of (5x+1)/(25x^2+60x-13) ?

The integral of (5x+1)/(25x^2+60x-13) is 1/35 ln|5x-1|+6/35 ln|5x+13|+C

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integral of (5x+1)/(25x^2+60x-13)

\int \frac{5 x + 1}{25 x ^{2} + 60 x - 13} d x

\frac{1}{35} ln ∣ 5 x - 1 ∣ + \frac{6}{35} ln ∣ 5 x + 13 ∣ + C

Solution steps

\int \frac{5 x + 1}{25 x ^{2} + 60 x - 13} d x

Take the partial fraction of

\frac{5 x + 1}{25 x ^{2} + 60 x - 13} : \frac{1}{7 ( 5 x - 1 )} + \frac{6}{7 ( 5 x + 13 )}

= \int \frac{1}{7 ( 5 x - 1 )} + \frac{6}{7 ( 5 x + 13 )} 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}{7 ( 5 x - 1 )} d x + \int \frac{6}{7 ( 5 x + 13 )} d x

\int \frac{1}{7 ( 5 x - 1 )} d x = \frac{1}{35} ln ∣ 5 x - 1 ∣

\int \frac{6}{7 ( 5 x + 13 )} d x = \frac{6}{35} ln ∣ 5 x + 13 ∣

= \frac{1}{35} ln ∣ 5 x - 1 ∣ + \frac{6}{35} ln ∣ 5 x + 13 ∣

Add a constant to the solution

= \frac{1}{35} ln ∣ 5 x - 1 ∣ + \frac{6}{35} ln ∣ 5 x + 13 ∣ + C

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What is the integral of (5x+1)/(25x^2+60x-13) ?
The integral of (5x+1)/(25x^2+60x-13) is 1/35 ln|5x-1|+6/35 ln|5x+13|+C