What is the integral of x/(27x^2-1) ?

The integral of x/(27x^2-1) is 1/54 ln|27x^2-1|+C

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integral of x/(27x^2-1)

\int \frac{x}{27 x ^{2} - 1} d x

\frac{1}{54} ln 27 x^{2} - 1 + C

Solution steps

\int \frac{x}{27 x ^{2} - 1} d x

Apply u-substitution

: \int \frac{1}{54 u} d u

= \int \frac{1}{54 u} d u

Take the constant out:

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

= \frac{1}{54} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{54} ln ∣ u ∣

Substitute back

u = 27 x^{2} - 1

= \frac{1}{54} ln 27 x^{2} - 1

Add a constant to the solution

= \frac{1}{54} ln 27 x^{2} - 1 + C

x \to 0 + lim (- x + 3)

\frac{d ^{2} y}{d x ^{2}} + 25 y = 0

x \to 1 - lim (\frac{1 - x}{1 + x})

\frac{8 x ^{2} + 8 x + 6}{x}

\frac{d}{d x} (x^{0.15})

What is the integral of x/(27x^2-1) ?
The integral of x/(27x^2-1) is 1/54 ln|27x^2-1|+C