What is the integral of 1/(51-70x-25x^2) ?

The integral of 1/(51-70x-25x^2) is -1/100 ln|5x-3|+1/100 ln|5x+17|+C

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integral of 1/(51-70x-25x^2)

\int \frac{1}{51 - 70 x - 25 x ^{2}} d x

- \frac{1}{100} ln ∣ 5 x - 3 ∣ + \frac{1}{100} ln ∣ 5 x + 17 ∣ + C

Solution steps

\int \frac{1}{51 - 70 x - 25 x ^{2}} d x

Take the partial fraction of

\frac{1}{51 - 70 x - 25 x ^{2}} : - \frac{1}{20 ( 5 x - 3 )} + \frac{1}{20 ( 5 x + 17 )}

= \int - \frac{1}{20 ( 5 x - 3 )} + \frac{1}{20 ( 5 x + 17 )} 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}{20 ( 5 x - 3 )} d x + \int \frac{1}{20 ( 5 x + 17 )} d x

\int \frac{1}{20 ( 5 x - 3 )} d x = \frac{1}{100} ln ∣ 5 x - 3 ∣

\int \frac{1}{20 ( 5 x + 17 )} d x = \frac{1}{100} ln ∣ 5 x + 17 ∣

= - \frac{1}{100} ln ∣ 5 x - 3 ∣ + \frac{1}{100} ln ∣ 5 x + 17 ∣

Add a constant to the solution

= - \frac{1}{100} ln ∣ 5 x - 3 ∣ + \frac{1}{100} ln ∣ 5 x + 17 ∣ + C

What is the integral of 1/(51-70x-25x^2) ?
The integral of 1/(51-70x-25x^2) is -1/100 ln|5x-3|+1/100 ln|5x+17|+C