What is the integral of (x^2)/(x^3-5x) ?

The integral of (x^2)/(x^3-5x) is 1/2 ln(|x+sqrt(5)||x-sqrt(5)|)+C

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integral of (x^2)/(x^3-5x)

\int \frac{x ^{2}}{x ^{3} - 5 x} d x

\frac{1}{2} ln (x + 5 x - 5) + C

Solution steps

\int \frac{x ^{2}}{x ^{3} - 5 x} d x

Factor

\frac{x ^{2}}{x ^{3} - 5 x}

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

Cancel the common factor:

x

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

Apply u-substitution

: \int \frac{1}{2 u} d u

= \int \frac{1}{2 u} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} ln ∣ u ∣

Substitute back

u = (x + 5) (x - 5)

= \frac{1}{2} ln (x + 5) (x - 5)

Simplify

\frac{1}{2} ln (x + 5) (x - 5) : \frac{1}{2} ln (x + 5 x - 5)

= \frac{1}{2} ln (x + 5 x - 5)

Add a constant to the solution

= \frac{1}{2} ln (x + 5 x - 5) + C

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