What is the integral of (x+1)/(x^3+3x^2-10x) ?

The integral of (x+1)/(x^3+3x^2-10x) is -1/10 ln|x|+3/14 ln|x-2|-4/35 ln|x+5|+C

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integral of (x+1)/(x^3+3x^2-10x)

\int \frac{x + 1}{x ^{3} + 3 x ^{2} - 10 x} d x

- \frac{1}{10} ln ∣ x ∣ + \frac{3}{14} ln ∣ x - 2 ∣ - \frac{4}{35} ln ∣ x + 5 ∣ + C

Solution steps

\int \frac{x + 1}{x ^{3} + 3 x ^{2} - 10 x} d x

Take the partial fraction of

\frac{x + 1}{x ^{3} + 3 x ^{2} - 10 x} : - \frac{1}{10 x} + \frac{3}{14 ( x - 2 )} - \frac{4}{35 ( x + 5 )}

= \int - \frac{1}{10 x} + \frac{3}{14 ( x - 2 )} - \frac{4}{35 ( x + 5 )} 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} d x + \int \frac{3}{14 ( x - 2 )} d x - \int \frac{4}{35 ( x + 5 )} d x

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

\int \frac{3}{14 ( x - 2 )} d x = \frac{3}{14} ln ∣ x - 2 ∣

\int \frac{4}{35 ( x + 5 )} d x = \frac{4}{35} ln ∣ x + 5 ∣

= - \frac{1}{10} ln ∣ x ∣ + \frac{3}{14} ln ∣ x - 2 ∣ - \frac{4}{35} ln ∣ x + 5 ∣

Add a constant to the solution

= - \frac{1}{10} ln ∣ x ∣ + \frac{3}{14} ln ∣ x - 2 ∣ - \frac{4}{35} ln ∣ x + 5 ∣ + C

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What is the integral of (x+1)/(x^3+3x^2-10x) ?
The integral of (x+1)/(x^3+3x^2-10x) is -1/10 ln|x|+3/14 ln|x-2|-4/35 ln|x+5|+C