What is the integral of (x-3)/(x^2-49) ?

The integral of (x-3)/(x^2-49) is 5/7 ln|x+7|+2/7 ln|x-7|+C

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integral of (x-3)/(x^2-49)

\int \frac{x - 3}{x ^{2} - 49} d x

\frac{5}{7} ln ∣ x + 7 ∣ + \frac{2}{7} ln ∣ x - 7 ∣ + C

Solution steps

\int \frac{x - 3}{x ^{2} - 49} d x

Take the partial fraction of

\frac{x - 3}{x ^{2} - 49} : \frac{5}{7 ( x + 7 )} + \frac{2}{7 ( x - 7 )}

= \int \frac{5}{7 ( x + 7 )} + \frac{2}{7 ( x - 7 )} 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{5}{7 ( x + 7 )} d x + \int \frac{2}{7 ( x - 7 )} d x

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

\int \frac{2}{7 ( x - 7 )} d x = \frac{2}{7} ln ∣ x - 7 ∣

= \frac{5}{7} ln ∣ x + 7 ∣ + \frac{2}{7} ln ∣ x - 7 ∣

Add a constant to the solution

= \frac{5}{7} ln ∣ x + 7 ∣ + \frac{2}{7} ln ∣ x - 7 ∣ + C

\int - ln ∣ x ∣ d x

\int_{0}^{1} \int x e^{x} d x d x

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

\frac{d}{d x} (x + \frac{25}{x})

(ln (\frac{x ( x ^{2} + 1 ) ^{2}}{2 x ^{3} - 1}))^{^{'}}

What is the integral of (x-3)/(x^2-49) ?
The integral of (x-3)/(x^2-49) is 5/7 ln|x+7|+2/7 ln|x-7|+C