What is the integral of ((5-10x^2)/(4x^3-6x+15)) ?

The integral of ((5-10x^2)/(4x^3-6x+15)) is -5/6 ln|4x^3-6x+15|+C

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integral of ((5-10x^2)/(4x^3-6x+15))

\int (\frac{5 - 10 x ^{2}}{4 x ^{3} - 6 x + 15}) d x

- \frac{5}{6} ln 4 x^{3} - 6 x + 15 + C

Solution steps

\int \frac{5 - 10 x ^{2}}{4 x ^{3} - 6 x + 15} d x

Apply u-substitution

: \int - \frac{5}{6 u} d u

= \int - \frac{5}{6 u} d u

Take the constant out:

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

= - \frac{5}{6} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{5}{6} ln ∣ u ∣

Substitute back

u = 4 x^{3} - 6 x + 15

= - \frac{5}{6} ln 4 x^{3} - 6 x + 15

Add a constant to the solution

= - \frac{5}{6} ln 4 x^{3} - 6 x + 15 + C

f (x) = x x^{2} + 21

x \to \infty lim (1 + 4 x)

x^{2} \frac{d y}{d x} - x y + 1 = 0

x \to \infty lim (- \frac{x}{x})

\int_{0}^{3} 2 e^{x} d x

What is the integral of ((5-10x^2)/(4x^3-6x+15)) ?
The integral of ((5-10x^2)/(4x^3-6x+15)) is -5/6 ln|4x^3-6x+15|+C