What is the integral of 1/(((x-3)(x+7))) ?

The integral of 1/(((x-3)(x+7))) is 1/10 ln|x-3|-1/10 ln|x+7|+C

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integral of 1/(((x-3)(x+7)))

integral

\frac{1}{( ( x - 3 ) ( x + 7 ) )}

\frac{1}{10} ln ∣ x - 3 ∣ - \frac{1}{10} ln ∣ x + 7 ∣ + C

Solution steps

\int \frac{1}{( x - 3 ) ( x + 7 )} d x

Take the partial fraction of

\frac{1}{( x - 3 ) ( x + 7 )} : \frac{1}{10 ( x - 3 )} - \frac{1}{10 ( x + 7 )}

= \int \frac{1}{10 ( x - 3 )} - \frac{1}{10 ( 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{1}{10 ( x - 3 )} d x - \int \frac{1}{10 ( x + 7 )} d x

\int \frac{1}{10 ( x - 3 )} d x = \frac{1}{10} ln ∣ x - 3 ∣

\int \frac{1}{10 ( x + 7 )} d x = \frac{1}{10} ln ∣ x + 7 ∣

= \frac{1}{10} ln ∣ x - 3 ∣ - \frac{1}{10} ln ∣ x + 7 ∣

Add a constant to the solution

= \frac{1}{10} ln ∣ x - 3 ∣ - \frac{1}{10} ln ∣ x + 7 ∣ + C

- 5 cos (\frac{x}{2})

e^{(x^{2}) - x}

\frac{1}{( x ^{2} + a ) ^{2}}

\frac{sin ^{2} ( 2 x )}{cos ^{2} ( x )}

e^{\frac{- x ^{2}}{3}}

What is the integral of 1/(((x-3)(x+7))) ?
The integral of 1/(((x-3)(x+7))) is 1/10 ln|x-3|-1/10 ln|x+7|+C