What is the integral of (x^9)/(x^{10)-1} ?

The integral of (x^9)/(x^{10)-1} is 1/10 ln|x^{10}-1|+C

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integral of (x^9)/(x^{10)-1}

\int \frac{x ^{9}}{x ^{10} - 1} d x

\frac{1}{10} ln x^{10} - 1 + C

Solution steps

\int \frac{x ^{9}}{x ^{10} - 1} d x

Apply u-substitution

= \int \frac{1}{10 u} d u

Take the constant out:

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

= \frac{1}{10} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{10} ln ∣ u ∣

Substitute back

u = x^{10} - 1

= \frac{1}{10} ln x^{10} - 1

Add a constant to the solution

= \frac{1}{10} ln x^{10} - 1 + C

What is the integral of (x^9)/(x^{10)-1} ?
The integral of (x^9)/(x^{10)-1} is 1/10 ln|x^{10}-1|+C