What is the integral of (x^4-3x^2)/(x^5-5x^3-24) ?

The integral of (x^4-3x^2)/(x^5-5x^3-24) is 1/5 ln|x^5-5x^3-24|+C

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integral of (x^4-3x^2)/(x^5-5x^3-24)

\int \frac{x ^{4} - 3 x ^{2}}{x ^{5} - 5 x ^{3} - 24} d x

\frac{1}{5} ln x^{5} - 5 x^{3} - 24 + C

Solution steps

\int \frac{x ^{4} - 3 x ^{2}}{x ^{5} - 5 x ^{3} - 24} d x

Apply u-substitution

: \int \frac{1}{5 u} d u

= \int \frac{1}{5 u} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{5} ln ∣ u ∣

Substitute back

u = x^{5} - 5 x^{3} - 24

= \frac{1}{5} ln x^{5} - 5 x^{3} - 24

Add a constant to the solution

= \frac{1}{5} ln x^{5} - 5 x^{3} - 24 + C

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What is the integral of (x^4-3x^2)/(x^5-5x^3-24) ?
The integral of (x^4-3x^2)/(x^5-5x^3-24) is 1/5 ln|x^5-5x^3-24|+C