What is the integral of (x^4+9x^2+5)/(x^5+15x^3+25x) ?

The integral of (x^4+9x^2+5)/(x^5+15x^3+25x) is 1/5 ln|x^5+15x^3+25x|+C

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integral of (x^4+9x^2+5)/(x^5+15x^3+25x)

\int \frac{x ^{4} + 9 x ^{2} + 5}{x ^{5} + 15 x ^{3} + 25 x} d x

\frac{1}{5} ln x^{5} + 15 x^{3} + 25 x + C

Solution steps

\int \frac{x ^{4} + 9 x ^{2} + 5}{x ^{5} + 15 x ^{3} + 25 x} d x

Apply u-substitution

= \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} + 15 x^{3} + 25 x

= \frac{1}{5} ln x^{5} + 15 x^{3} + 25 x

Add a constant to the solution

= \frac{1}{5} ln x^{5} + 15 x^{3} + 25 x + C

What is the integral of (x^4+9x^2+5)/(x^5+15x^3+25x) ?
The integral of (x^4+9x^2+5)/(x^5+15x^3+25x) is 1/5 ln|x^5+15x^3+25x|+C