What is the integral of 5/(x^2-10x) ?

The integral of 5/(x^2-10x) is 5(-1/10 ln|x|+1/10 ln|x-10|)+C

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integral of 5/(x^2-10x)

\int \frac{5}{x ^{2} - 10 x} d x

5 (- \frac{1}{10} ln ∣ x ∣ + \frac{1}{10} ln ∣ x - 10 ∣) + C

Solution steps

\int \frac{5}{x ^{2} - 10 x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int \frac{1}{x ^{2} - 10 x} d x

Take the partial fraction of

\frac{1}{x ^{2} - 10 x} : - \frac{1}{10 x} + \frac{1}{10 ( x - 10 )}

= 5 \cdot \int - \frac{1}{10 x} + \frac{1}{10 ( x - 10 )} 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

= 5 (- \int \frac{1}{10 x} d x + \int \frac{1}{10 ( x - 10 )} d x)

\int \frac{1}{10 x} d x = \frac{1}{10} ln ∣ x ∣

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

= 5 (- \frac{1}{10} ln ∣ x ∣ + \frac{1}{10} ln ∣ x - 10 ∣)

Add a constant to the solution

= 5 (- \frac{1}{10} ln ∣ x ∣ + \frac{1}{10} ln ∣ x - 10 ∣) + C

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What is the integral of 5/(x^2-10x) ?
The integral of 5/(x^2-10x) is 5(-1/10 ln|x|+1/10 ln|x-10|)+C